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Abstract and Applied Analysis
Volume 2013, Article ID 202851, 6 pages
http://dx.doi.org/10.1155/2013/202851
Research Article

Simplicity and Spectrum of Singular Hamiltonian Systems of Arbitrary Order

Department of Mathematics, Shandong University at Weihai, Weihai, Shandong 264209, China

Received 22 August 2013; Accepted 5 December 2013

Academic Editor: Douglas Anderson

Copyright © 2013 Huaqing Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. V. I. Kogan and F. S. Rofe-Beketov, “On square-integrable solutions of symmetric systems of differential equations of arbitrary order,” Proceedings of the Royal Society of Edinburgh, vol. 74, pp. 5–40, 1976. View at Google Scholar · View at MathSciNet
  2. P. W. Walker, “A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square,” Journal of the London Mathematical Society, vol. 9, pp. 151–159, 1974/75. View at Google Scholar · View at MathSciNet
  3. F. V. Atkinson, Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1964. View at MathSciNet
  4. R. Anderson, “Continuous spectra of a singular symmetric differential operator on a Hilbert space of vector-valued functions,” Pacific Journal of Mathematics, vol. 55, pp. 1–7, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. H. Behncke and D. B. Hinton, “Eigenfunctions, deficiency indices and spectra of odd-order differential operators,” Proceedings of the London Mathematical Society, vol. 97, no. 2, pp. 425–449, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. H. Behncke and D. Hinton, “Spectral theory of Hamiltonian systems with almost constant coefficients,” Journal of Differential Equations, vol. 250, no. 3, pp. 1408–1426, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. H. Behncke and F. O. Nyamwala, “Spectral analysis of higher order differential operators with unbounded coefficients,” Mathematische Nachrichten, vol. 285, no. 1, pp. 56–73, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. X. Hao, J. Sun, and A. Zettl, “Real-parameter square-integrable solutions and the spectrum of differential operators,” Journal of Mathematical Analysis and Applications, vol. 376, no. 2, pp. 696–712, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. X. Hao, J. Sun, and A. Zettl, “The spectrum of differential operators and square-integrable solutions,” Journal of Functional Analysis, vol. 262, no. 4, pp. 1630–1644, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. C. Remling, “Spectral analysis of higher order differential operators. I. General properties of the m-function,” Journal of the London Mathematical Society, vol. 58, no. 2, pp. 367–380, 1998. View at Publisher · View at Google Scholar · View at MathSciNet
  11. C. Remling, “Spectral analysis of higher-order differential operators. II. Fourth-order equations,” Journal of the London Mathematical Society, vol. 59, no. 1, pp. 188–206, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. J. Sun, A. Wang, and A. Zettl, “Continuous spectrum and square-integrable solutions of differential operators with intermediate deficiency index,” Journal of Functional Analysis, vol. 255, no. 11, pp. 3229–3248, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. J. Weidmann, Spectral Theory of Ordinary Differential Operators, vol. 1258 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1987. View at MathSciNet
  14. P. Hartman and A. Wintner, “An oscillation theorem for continuous spectra,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 376–379, 1947. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. P. Hartman and A. Wintner, “A separation theorem for continuous spectra,” American Journal of Mathematics, vol. 71, pp. 650–662, 1949. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. P. Hartman and A. Wintner, “On the essential spectra of singular eigenvalue problems,” American Journal of Mathematics, vol. 72, pp. 545–552, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. Y. Shi, “On the rank of the matrix radius of the limiting set for a singular linear Hamiltonian system,” Linear Algebra and its Applications, vol. 376, pp. 109–123, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in Hilbert Space II, Pitman, London, UK, 1981.
  19. R. C. Gilbert, “Simplicity of linear ordinary differential operators,” Journal of Differential Equations, vol. 11, pp. 672–681, 1972. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. V. Mogilevskii, “Symmetric operators with real defect subspaces of the maximal dimension. Applications to differential operators,” Journal of Functional Analysis, vol. 261, no. 7, pp. 1955–1968, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. J. Weidmann, Linear Operators in Hilbert Spaces, vol. 68 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1980, Translated from the German by Joseph Szücs. View at MathSciNet
  22. J. Behrndt, S. Hassi, H. de Snoo, and R. Wietsma, “Square-integrable solutions and Weyl functions for singular canonical systems,” Mathematische Nachrichten, vol. 284, no. 11-12, pp. 1334–1384, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. H. Schmid and C. Tretter, “Singular Dirac systems and Sturm-Liouville problems nonlinear in the spectral parameter,” Journal of Differential Equations, vol. 181, no. 2, pp. 511–542, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet