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Essential and Absolute Spectra

  • Todd Kapitula
  • Keith Promislow
Chapter
Part of the Applied Mathematical Sciences book series (AMS, volume 185)

Abstract

The goal of this chapter is the characterization of the essential spectrum and Fredholm indices of two classes of linear differential operators on unbounded domains. The first class is comprised of nth-order differential operators with spatially varying coefficients that tend at an exponential rate to constant values at \(\pm \infty \).

Keywords

Essential Spectrum Morse Index Exponential Dichotomy Linear Partial Differential Equation Asymptotic Operator 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. [30]
    A. Ben-Artzi and I. Gohberg. Dichotomy of systems and invertibility of linear ordinary differential operators. Operator Theory Adv. Appl., 56: 91–119, 1992.Google Scholar
  2. [53]
    C. Chicone and Y. Latushkin. Evolution Semigroups in Dynamical Systems and Differential Equations, volume 70 of Math. Surv. Monogr. American Mathematical Society, Providence, RI, 1999.Google Scholar
  3. [59]
    W.A. Coppel. Dichotomies in stability theory. In Lecture Notes in Mathematics 629. Springer-Verlag, New York, 1978.Google Scholar
  4. [65]
    J. Daleckii and M. Krein. Stability of Solutions of Differential Equations in Banach Space, volume 43 of Trans. Math. Monogr. American Mathematical Society, Providence, RI, 1974.Google Scholar
  5. [82]
    B. Fiedler and A. Scheel. Spatio-Temporal Dynamics of Reaction–Diffusion Patterns, Trends in Nonlinear Analysis. Springer-Verlag, Berlin, 2003.Google Scholar
  6. [112]
    J. Hale. Ordinary Differential Equations. Robert E. Krieger Publishing, Malabar, FL, second edition, 1980.MATHGoogle Scholar
  7. [113]
    J. Härterich, B. Sandstede, and A. Scheel. Exponential dichotomies for linear non-autonomous functional differential equations of mixed type. Indiana U. Math. J., 51:1081–1109, 2002.MATHCrossRefGoogle Scholar
  8. [182]
    Y. Latushkin and A. Pogan. The dichotomy theorem for evolution bi-families. J. Diff. Eq., 245:2267–2306, 2008.MathSciNetMATHCrossRefGoogle Scholar
  9. [200]
    W. Magnus and S. Winkler. Hill’s Equation, volume 20 of Interscience Tracts in Pure and Applied Mathematics. Interscience, New York, 1966.Google Scholar
  10. [215]
    K. Palmer. Exponential dichomoties and transversal homoclinic points. J. Diff. Eq., 55(2):225–256, 1984.MathSciNetMATHCrossRefGoogle Scholar
  11. [216]
    K. Palmer. Exponential dichotomies and Fredholm operators. Proc. Amer. Math. Soc., 104(1):149–156, 1988.MathSciNetMATHCrossRefGoogle Scholar
  12. [221]
    R. Pego and M. Weinstein. Asymptotic stability of solitary waves. Comm. Math. Phys., 164:305–349, 1994.MathSciNetMATHCrossRefGoogle Scholar
  13. [239]
    J. Rademacher, B. Sandstede, and A. Scheel. Computing absolute and essential spectra using continuation. Physica D, 229(1&2):166–183, 2007.MathSciNetMATHCrossRefGoogle Scholar
  14. [242]
    M. Reed and B. Simon. Methods of Modern Mathematical Physics IV: Analysis of Operators. Academic Press, New York, 1978.MATHGoogle Scholar
  15. [250]
    B. Sandstede. Stability of travelling waves. In Handbook of Dynamical Systems, volume 2, chapter 18, pp. 983–1055. Elsevier, New York, 2002.Google Scholar
  16. [252]
    B. Sandstede and A. Scheel. Absolute and convective instabilities of waves on unbounded and large bounded domains. Physica D, 145:233–277, 2000a.MathSciNetMATHCrossRefGoogle Scholar
  17. [255]
    B. Sandstede and A. Scheel. On the structure of spectra of modulated travelling waves. Math. Nachr., 232:39–93, 2001b.MathSciNetMATHCrossRefGoogle Scholar
  18. [257]
    B. Sandstede and A. Scheel. Relative Morse indices, Fredholm indices, and group velocities. Disc. Cont. Dyn. Syst., 20:139–158, 2008.MathSciNetMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Todd Kapitula
    • 1
  • Keith Promislow
    • 2
  1. 1.Department of Mathematics and StatisticsCalvin CollegeGrand RapidsUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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