Skip to main content
SpringerLink
Log in
Book cover

Differential Operators and Related Topics pp 393–417Cite as

Birkhäuser

On M.G. Krein’s Spectral Shift Function for Canonical Systems of Differential Equations

  • Conference paper

  • 338 Accesses

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

Abstract

In this paper the spectral shift function (s.s.f.) for the pair of operators, generated by a canonical system of differential equations with two different boundary conditions, is studied. The main result is a connection between the s.s.f. and the spectral function. Also the inverse problem is studied: For a given s.s.f. we find a canonical system and investigate its uniqueness.

This research was made possible in Part by Grant no.93–0249 from INTAS

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. N.I. Ahiezer and I.M. Glazman, The theory of linear operators in Hilbert space (Russian), Nauka, Moscow, 1961.

    Google Scholar 

  2. F.V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York, 1964.

    MATH  Google Scholar 

  3. E.A. Coddington and N. Levinson, Theory of ordinary differential equations, New York, 1955.

    MATH  Google Scholar 

  4. L. de Branges, Some Hilbert spaces of entire functions. III, Trans. Amer. Math. Soc. 100 (1961), 73–115.

    Article  MathSciNet  MATH  Google Scholar 

  5. L. de Branges, Some Hilbert spaces of entire functions. II, Trans. Amer. Math. Soc. 99 (1961), 118–152.

    Article  MathSciNet  MATH  Google Scholar 

  6. L. de Branges, Some Hilbert spaces of entire functions. IV, Trans. Amer. Math. Soc. 105 (1962), 43–83.

    Article  MathSciNet  MATH  Google Scholar 

  7. I.C. Gohberg and M.G. Krein, Introduction in the theory of linear nonself-adjoint operators (Russian), Nauka, Moscow, 1965.

    Google Scholar 

  8. I.C. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert spaces (Russian), Nauka, Moscow, 1967.

    Google Scholar 

  9. I.S. Kac, Linear relations, generated by a canonical differential equation (Russian), Functional Anal. i Priložen. 17 (1983), 86–87.

    Google Scholar 

  10. I.S. Kac, Linear relations, generated by a canonical differential equation on an interval with a regular endpoint and expansibility in eigenfunctions (Russian), Odessa, 1984.

    Google Scholar 

  11. I.S. Kac and M.G. Krein, R-functions-analytic functions mapping the upper half-plane into itself, Amer. Math. Soc. Transl. (2) 103 (1974), 1–17.

    MATH  Google Scholar 

  12. M.G. Krein, On the trace formula in perturbation theory (Russian), Math. Sb. 33 (1953), 597–626.

    MathSciNet  Google Scholar 

  13. M.G. Krein, On the perturbation determinant and the trace formula for uni-tary and self-adjoint operators (Russian), Dokl. Akad. Nauk SSSR 144 (1962), 268–271.

    MathSciNet  Google Scholar 

  14. M.G. Krein, About some new investigations in perturbation theory for self-adjoint operators (Russian), First Summer School of mathematics, Kanev, 1963.

    Google Scholar 

  15. M.G. Krein and V.A. Yavrian, On the spectral shift function arising from the perturbation of a positive operator (Russian), J. Operator Theory 6 (1981), 155–191.

    MathSciNet  Google Scholar 

  16. H. Langer and B. Textorius, A generalization of M.G. Krein’s method of directing functionals to linear relations, Proc. Roy. Soc. Edinburgh 81A (1978), 237–246.

    Article  MathSciNet  Google Scholar 

  17. B.M. Levitan and Y.S. Gasimov, Determination of the differential equation by two spectrums (Russian), Usp. matem. nauk. 19 (1964), 3–63.

    MATH  Google Scholar 

  18. B.M. Levitan and I.S. Sargsjan, Introduction to spectral theory of self-adjoint ordinary differential operators, Trans. of Math. Monographs 39 (Amer. Math. Soc., Providence, 1975).

    Google Scholar 

  19. V.A. Marchenko, Spectral theory of Sturm-Lioville operators (Russian), “Naukova dumka”, Kiev, 1972.

    Google Scholar 

  20. V.A. Yavrian, The asymptotics of the Weyl function for canonical system of dif-ferential equations, Journal of Contemporary Mathematical Analysis (Armenian Academy of Sciences), vol. 33, no. 4,1998.

    Google Scholar 

  21. V.A. Javrian (Yavrian), On the regularized trace of the difference between two singular Sturm-Lioville operators, Soviei. Math. Dokl. 7 (1966), 888–891.

    Google Scholar 

  22. V.A. Yavrian, On the asymptotics of the spectral matrix-function of canonical system of differential equations (Russian), Dokl. Akad. Nauk Armenian SSR 56 (1973), 129–134.

    Google Scholar 

  23. H. Winkler, The inverse spectral problem for canonical systems, Integr. Equat. Oper. Th. 22 (1995), 360–374.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Yerevan State University, Alek Manukian str. 1, Yerevan, 375049, Armenia

    V. A. Yavrian

Authors
  1. V. A. Yavrian
    View author publications

    You can also search for this author in PubMed Google Scholar

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

Rights and permissions

Reprints and permissions

Copyright information

© 2000 Springer Basel AG

About this paper

Cite this paper

Yavrian, V.A. (2000). On M.G. Krein’s Spectral Shift Function for Canonical Systems of Differential Equations. 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_30

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8403-7_30

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9552-1

  • Online ISBN: 978-3-0348-8403-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation