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Ukrainian Mathematical Journal

, Volume 71, Issue 6, pp 977–983 | Cite as

Instability Intervals for Hill’s Equation with Symmetric Single-Well Potential

  • H. Coşkun
  • E. BaşkayaEmail author
  • A. Kabataş
Article

With the help of an auxiliary eigenvalue problem, we deduce some explicit estimates for the periodic and semiperiodic eigenvalues and the lengths of instability intervals for Hill’s equation with symmetric single-well potentials. We also establish bounds for the gaps in the sets of Dirichlet and Neumann eigenvalues.

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References

  1. 1.
    H. Coşkun and B. J. Harris, “Estimates for the periodic and semiperiodic eigenvalues of Hill’s equations,” Proc. Roy. Soc. Edinburgh Sect. A, 130, 991–998 (2000).MathSciNetCrossRefGoogle Scholar
  2. 2.
    H. Coşkun, “Some inverse results for Hill’s equation,” J. Math. Anal. Appl., 276, 833–844 (2002).MathSciNetCrossRefGoogle Scholar
  3. 3.
    H. Coşkun, “On the spectrum of a second order periodic differential equation,” Rocky Mountain J. Math., 33, 1261–1277 (2003).MathSciNetCrossRefGoogle Scholar
  4. 4.
    M. S. P. Eastham, The Spectral Theory of Periodic Differential Equations, Scottish Academic Press, Edinburgh; London (1973).zbMATHGoogle Scholar
  5. 5.
    N. B. Haaser and J. A. Sullivan, Real Analysis, Van Nostrand Reinhold Co., New York (1991).zbMATHGoogle Scholar
  6. 6.
    H. Hochstadt, “On the determination of a Hill’s equation from its spectrum,” Arch. Ration. Mech. Anal., 19, 353–362 (1965).MathSciNetCrossRefGoogle Scholar
  7. 7.
    M. J. Huang, “The first instability interval for Hill equations with symmetric single well potentials,” Proc. Amer. Math. Soc., 125, 775–778 (1997).MathSciNetCrossRefGoogle Scholar
  8. 8.
    M. J. Huang and T. M. Tsai, “The eigenvalue gap for one-dimensional Schrödinger operators with symmetric potentials,” Proc. Roy. Soc. Edinburgh Sect. A, 139, 359–366 (2009).MathSciNetCrossRefGoogle Scholar
  9. 9.
    A. Ntinos, “Lengths of instability intervals of second order periodic differential equations,” Q. J. Math., 27, 387–394 (1976).MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Karadeniz Technical UniversityTrabzonTurkey

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