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Differential Operator Matrices of Mixed Order with Periodic Coefficients

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Differential Operators and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 117))

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Abstract

We study spectral properties of 2 x 2 operator matrices H defined in the Hilbert space ℍ = L 2(R) x L 2(R)by linear differential systems of mixed order with periodic coefficients. We prove that the spectrum σ (H) of H has a band and gap structure and consists of two band sequences one of which, when infinite, has a finite accumulation point, and give sufficient conditions for this accumulation to take place.

This work was supported by Russian Foundation for Fundamental Research (RFFI), grant No.98–01–01000 and grant No. 96–15–96091

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References

  1. V.M. Adamyan and H. Langer, Spectral properties of a class of rational oper-ator valued functions,J. Operator Theory 33 (1995), 259–277.

    MathSciNet  Google Scholar 

  2. V. Adamyan, H. Langer, R. Mennicken and J. SaurerSpectral components of selfadjoint block operator matrices with unbounded entries, Math. Nachr. 178 (1996), 43–80.

    Article  MathSciNet  MATH  Google Scholar 

  3. F.V. Atkinson, H. Langer, R. Mennicken and A.A. ShkalikovThe essential spectrum of some matrix operators, Math. Nachr. 167 (1994), 520.

    Article  MathSciNet  Google Scholar 

  4. S. Agmon, A. Douglis and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Comm. Pure Appl. Math. 12 (1959), 623–727 and II, 17 (1964), 35–92.

    Google Scholar 

  5. M. Faierman, R. Mennicken and M. Möller, The essential spectrum of singular ordinary differential operators of mixed order. I, II, Math. Nachr. (to appear).

    Google Scholar 

  6. G. Grubb and G. Geymonat, The essential spectrum of elliptic systems of mixed order, Math. Ann. 227 (1977), 247–276.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Hardt, R. Mennicken and S.N. Naboko, Systems of singular differential operators of mixed order and applications to one-dimensional MHD problem, Math. Nachr. (to appear).

    Google Scholar 

  8. P. Hartman, Ordinary Differential Equations. John Wiley & Sons: New York,1964.

    MATH  Google Scholar 

  9. R.O. Hryniv and T.A. Melnyk, On a singular Rayleighfunctional, Mat. Zametki 60 (1996), no. 1,130–134 (Russian); translation in Math. Notes 60 (1996), no. 1–2,97–100.

    Google Scholar 

  10. R.O. Hryniv, A.A. Shkalikov and A.A. Vladimirov, Spectral analysis of peri-odic differential operator matrices of mixed order (to appear).

    Google Scholar 

  11. T. Kato, Perturbation Theory of Linear Operators. Springer-Verlag, Berlin,1976.

    Book  Google Scholar 

  12. A.Yu. Konstantinov, Spectral theory of some matrix differential operators of mixed order, Universität Bielefeld, Preprint no. 97–060.

    Google Scholar 

  13. L.D. Landau and E.M. Lifshitz, Fluid Mechanics. London: Pergamon Press, 1959.

    Google Scholar 

  14. H. Langer, R. Mennicken and M. Möller, A second order differential operator depending non-linearly on the eigenvalue parameter. Oper. Theory Adv. Appl. 48 (1990), 319–332.

    Google Scholar 

  15. A.E. Lifschitz, Magnetohydrodynamics and Spectral Theory. Kluwer Acad. Publishers: Dodrecht, 1989.

    Book  MATH  Google Scholar 

  16. R. Mennicken, H. Schmid and A.A. Shkalikov, On the eigenvalue accumulation of Sturm-Liouville problem depending nonlinearly on the spectral parameter,Math. Nachr. 189 (1998), 157–170.

    Article  MathSciNet  MATH  Google Scholar 

  17. R. Mennicken and A.A. Shkalikov, Spectral decomposition of operator matrices, Math Nachr. 179 (1996), 259–273.

    Article  MathSciNet  MATH  Google Scholar 

  18. A.K. Motovilov, The removal of an energy dependence from the interaction in two body systems, J. Math. Phys. 32 (1991), 3509–3518.

    Article  MathSciNet  Google Scholar 

  19. M. Reed and B.Simon, Methods of Modern Mathematical Physics, vol. IV. Academic Press: London, 1978.

    Google Scholar 

  20. A.A. Shkalikov, On essential spectrum of matrix operators,Math. Notes 58 (1995), no. 6,945–949.

    Article  MathSciNet  Google Scholar 

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Author information

Author notes

  1. R. Hryniv

    Present address: Department of Mathematics and Statistics, University of Calgary, Calgary, AB, T2N 1N4, Canada

Authors and Affiliations

  1. Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova str., Lviv, 290601, Ukraine

    R. Hryniv

  2. Department of Mechanics and Mathematics, Moscow State University, Moscow, 119899, Russia

    A. Shkalikov

Authors
  1. R. Hryniv
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  2. A. Shkalikov
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  3. A. Vladimirov
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Editor information

Editors and Affiliations

  1. Department of Theoretical Physics, University of Odessa, 270026, Odessa, Ukraine

    V. M. Adamyan

  2. Department of Mathematical Sciences Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, 69978, Ramat Aviv, Israel

    I. Gohberg

  3. Institute of Mathematics, National Academy of Sciences of Ukraine, Kyiv, Ukraine

    M. Gorbachuk  & V. Gorbachuk  & 

  4. Department of Mathematics, Vrije Universiteit, De Boelelaan 1081a, 1081 HV, Amsterdam, The Netherlands

    M. A. Kaashoek

  5. Department of Mathematics, Technical University of Vienna, Wiedner Hauptstrasse 8-10/1411, 1040, Vienna, Austria

    H. Langer

  6. Institute of Mathematics, Economics and Mechanics, Odessa State University, 270057, Dvoryanskaya str. 2, Odessa, Ukraine

    G. Popov

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Hryniv, R., Shkalikov, A., Vladimirov, A. (2000). Differential Operator Matrices of Mixed Order with Periodic Coefficients. In: Adamyan, V.M., et al. Differential Operators and Related Topics. Operator Theory: Advances and Applications, vol 117. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8403-7_13

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  • DOI: https://doi.org/10.1007/978-3-0348-8403-7_13

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9552-1

  • Online ISBN: 978-3-0348-8403-7

  • eBook Packages: Springer Book Archive

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