Ukrainian Mathematical Journal

, Volume 42, Issue 10, pp 1228–1233 | Cite as

Splitting and the spectrum of a linear differential equation with quasiperiodic coefficients

  • V. I. Tkachenko


A linear differential equation with quasiperiodic coefficients for large values of the spectral parameter is split into an exponentially dichotomous system and a two-dimensional] system with the properties of the one-dimensional Schrödinger equation. The set of the values of the parameter, for which the equation has solution of the form A(t) epx (iat) with a real number a and a quasiperiodic function A(t), is a single out.


Differential Equation Real Number Spectral Parameter Linear Differential Equation Quasiperiodic Function 
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Literature cited

  1. 1.
    E. I. Dinaburg and Ya. G. Sinai, “On the one-dimensional Schrödinger equation with quasiperiodic potential,” Funkts. Anal. Prilozhen.,9, No. 4, 8–21 (1975).Google Scholar
  2. 2.
    I. O. Parasyuk, “On the instability zones of the Schrödinger equation with a smooth quasiperiodic potential,” Ukr. Mat. Zh.,30, No. 1, 70–78 (1978).Google Scholar
  3. 3.
    J. Moser and J. Pöschel, “An extension of a result of Dinaburg and Sinai on quasiperiodic potentials,” Comment. Math Helv.,59, No. 1, 39–85 (1984).Google Scholar
  4. 4.
    H. Russmann, “On the one-dimensional Schrödinger equation with a quasiperiodic potential,” Ann. New York Acad. Sci., No. 357, 90–107 (1980).Google Scholar
  5. 5.
    A. M. Samoilenko, Elements of the Mathematical theory of Multifrequency Oscillations, Invariant Tori [in Russian], Nauka, Moscow (1987).Google Scholar
  6. 6.
    B. V. Shabat, Introduction to Complex Analysis, Part 2, Functions of Several Variables [in Russian], Nauka, Moscow (1976).Google Scholar
  7. 7.
    L. Chierchia, “Absolutely continuous spectra of quasiperiodic Schrödinger operators,” J. Math. Phys.,28, No. 12, 2891–2898 (1987).Google Scholar

Copyright information

© Plenum Publishing Corporation 1991

Authors and Affiliations

  • V. I. Tkachenko
    • 1
  1. 1.Institute of MathematicsAcademy of Sciences of the Ukrainian SSRKiev

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