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Transactions of the American Mathematical Society

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Sturmian and spectral theory for discrete symplectic systems


Authors: Martin Bohner, Ondrej Dosly and Werner Kratz
Journal: Trans. Amer. Math. Soc. 361 (2009), 3109-3123
MSC (2000): Primary 39A12, 39A13, 34B24, 49K99
DOI: https://doi.org/10.1090/S0002-9947-08-04692-8
Published electronically: December 30, 2008
MathSciNet review: 2485420
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Abstract | References | Similar Articles | Additional Information

Abstract: We consider $ 2n\times 2n$ symplectic difference systems together with associated discrete quadratic functionals and eigenvalue problems. We establish Sturmian type comparison theorems for the numbers of focal points of conjoined bases of a pair of symplectic systems. Then, using this comparison result, we show that the numbers of focal points of two conjoined bases of one symplectic system differ by at most $ n$. In the last part of the paper we prove the Rayleigh principle for symplectic eigenvalue problems and we show that finite eigenvectors of such eigenvalue problems form a complete orthogonal basis in the space of admissible sequences.


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

  • 1. R. P. Agarwal, M. Bohner, S. R. Grace, and D. O'Regan.
    Discrete Oscillation Theory.
    Hindawi Publishing Corporation, 2005.
  • 2. C. D. Ahlbrandt and A. C. Peterson.
    Discrete Hamiltonian Systems: Difference Equations, Continued Fractions, and Riccati Equations, volume 16 of Kluwer Texts in the Mathematical Sciences.
    Kluwer Academic Publishers, Boston, 1996. MR 1423802 (98m:39043)
  • 3. A. Ben-Israel and T. N. E. Greville.
    Generalized Inverses: Theory and Applications.
    John Wiley & Sons, Inc., New York, 1974. MR 0396607 (53:469)
  • 4. M. Bohner.
    Controllability and disconjugacy for linear Hamiltonian difference systems.
    In S. Elaydi, J. Graef, G. Ladas, and A. Peterson, editors, Conference Proceedings of the First International Conference on Difference Equations, pages 65-77, San Antonio, 1994. Gordon and Breach. MR 1678645
  • 5. M. Bohner.
    Zur positivität diskreter quadratischer Funktionale.
    Ph.D. thesis, Universität Ulm, 1995.
    English Edition: On positivity of discrete quadratic functionals.
  • 6. M. Bohner.
    Linear Hamiltonian difference systems: disconjugacy and Jacobi-type conditions.
    J. Math. Anal. Appl., 199(3):804-826, 1996. MR 1386607 (97a:39003)
  • 7. M. Bohner.
    Discrete Sturmian theory.
    Math. Inequal. Appl., 1(3):375-383, 1998.
    Preprint in Ulmer Seminare 1. MR 1629392 (99e:39008)
  • 8. M. Bohner and O. Došlý.
    Disconjugacy and transformations for symplectic systems.
    Rocky Mountain J. Math., 27(3):707-743, 1997. MR 1490271 (99e:39007)
  • 9. M. Bohner, O. Došlý, and W. Kratz.
    An oscillation theorem for discrete eigenvalue problems.
    Rocky Mountain J. Math., 33(4):1233-1260, 2003. MR 2052485 (2005d:39083)
  • 10. M. Bohner, O. Došlý, and W. Kratz.
    Positive semidefiniteness of discrete quadratic functionals.
    Proc. Edinburgh Math. Soc., 46:627-636, 2003. MR 2013957 (2004j:39022)
  • 11. O. Došlý and W. Kratz.
    A Sturmian separation theorem for symplectic difference systems.
    J. Math. Anal. Appl., 325:333-341, 2007. MR 2273528 (2007j:39006)
  • 12. O. Došlý and W. Kratz.
    Oscillation theorems for symplectic difference systems.
    J. Difference Equ. Appl., 13(7):585-605, 2007. MR 2336808
  • 13. J. Elyseeva.
    A transformation for symplectic systems and the definition of a focal point.
    Comput. Math. Appl., 47(1):123-134, 2004. MR 2062731 (2005b:39004)
  • 14. K. Feng.
    The Hamiltonian way for computing Hamiltonian dynamics.
    In Applied and Industrial Mathematics (Venice, 1989), volume 56 of Math. Appl., pages 17-35. Kluwer Acad. Publ., Dordrecht, 1991. MR 1147188 (92m:58043)
  • 15. R. Hilscher.
    Reid roundabout theorem for symplectic dynamic systems on time scales.
    Appl. Math. Optim., 43(2):129-146, 2001. MR 1814591 (2002a:37118)
  • 16. R. Hilscher and V. Růžičková.
    Implicit Riccati equations and quadratic functionals for discrete symplectic systems.
    Int. J. Difference Equ., 1(1):135-154, 2006. MR 2296502
  • 17. W. Kratz.
    Quadratic Functionals in Variational Analysis and Control Theory, volume 6 of Mathematical Topics.
    Akademie Verlag, Berlin, 1995. MR 1334092 (96f:34002)
  • 18. W. Kratz.
    Discrete oscillation.
    J. Difference Equ. Appl., 9(1):135-147, 2003. MR 1958308 (2004b:39020)
  • 19. J. Qi and S. Chen.
    Lower bound for the spectrum and the presence of pure point spectrum of a singular discrete Hamiltonian system.
    J. Math. Anal. Appl., 295(2):539-556, 2004. MR 2072031 (2005e:47090)
  • 20. V. Růžičková.
    Discrete symplectic systems and definiteness of quadratic functionals.
    Ph.D. thesis, Masaryk University Brno, 2006.
  • 21. W. T. Reid.
    Oscillation criteria for self-adjoint differential systems.
    Trans. Amer. Math. Soc., 101:91-106, 1961. MR 0133518 (24:A3349)
  • 22. W. T. Reid.
    Ordinary Differential Equations.
    John Wiley & Sons, Inc., New York, 1971. MR 0273082 (42:7963)
  • 23. W. T. Reid.
    Sturmian Theory for Ordinary Differential Equations.
    Springer-Verlag, New York, 1980. MR 606199 (82f:34002)
  • 24. Y. Shi.
    Symplectic structure of discrete Hamiltonian systems.
    J. Math. Anal. Appl., 266(2):472-478, 2002. MR 1880519 (2002k:37093)
  • 25. Y. Shi.
    Spectral theory of discrete linear Hamiltonian systems.
    J. Math. Anal. Appl., 289(2):554-570, 2004. MR 2026925 (2005a:39039)
  • 26. Y. Wang, Y. Shi, and G. Ren.
    Transformations for complex discrete linear Hamiltonian and symplectic systems.
    Bull. Aust. Math. Soc., 75(2):179-191, 2007. MR 2312562 (2008c:39008)

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Additional Information

Martin Bohner
Affiliation: Department of Mathematics and Statistics and Department of Economics and Finance, Missouri University of Science and Technology, Rolla, Missouri 65401

Ondrej Dosly
Affiliation: Department of Mathematics and Statistics, Masaryk University, CZ-61137, Brno, Czech Republic

Werner Kratz
Affiliation: Institut für Angewandte Analysis, Universität Ulm, D-89069 Ulm, Germany

DOI: https://doi.org/10.1090/S0002-9947-08-04692-8
Keywords: Discrete symplectic system, discrete quadratic functional, Sturmian separation result, Sturmian comparison result, Rayleigh principle, extended Picone identity
Received by editor(s): June 20, 2007
Published electronically: December 30, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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