Skip to main content
SpringerLink
Log in

Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

We develop an analog of classical oscillation theory for Sturm–Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators.

This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the number of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein’s spectral shift function is established.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Birman M.Sh.: On the spectrum of singular boundary value problems. AMS Translations (2) 53, 23–80 (1966)

    MATH  Google Scholar 

  2. Eastham M.S.P.: The spectral theory of periodic differential equations. Scottish Academic Press, Edinburgh (1973)

    MATH  Google Scholar 

  3. Gesztesy F., Simon B.: A short proof of Zheludev’s theorem. Trans. Am. Math. Soc. 335, 329–340 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  4. Gesztesy F., Ünal M.: Perturbative oscillation criteria and Hardy-type inequalities. Math. Nachr. 189, 121–144 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  5. Gesztesy F., Simon B., Teschl G.: Zeros of the Wronskian and renormalized oscillation Theory. Am. J. Math. 118, 571–594 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  6. Gohberg I., Goldberg S., Krupnik N.: Traces and Determinants of Linear Operators. Birkhäuser, Basel (2000)

    MATH  Google Scholar 

  7. Hartman P.: Differential equations with non-oscillatory eigenfunctions. Duke Math. J. 15, 697–709 (1948)

    Article  MathSciNet  Google Scholar 

  8. Hartman P.: A characterization of the spectra of one-dimensional wave equations. Am. J. Math. 71, 915–920 (1949)

    Article  MATH  MathSciNet  Google Scholar 

  9. Hartman P., Putnam C.R.: The least cluster point of the spectrum of boundary value problems. Am. J. Math. 70, 849–855 (1948)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kato T.: Perturbation Theory for Linear Operators. Springer, New York (1966)

    MATH  Google Scholar 

  11. Kneser A.: Untersuchungen über die reellen Nullstellen der Integrale linearer Differentialgleichungen. Math. Ann. 42, 409–435 (1893)

    Article  MathSciNet  Google Scholar 

  12. Khryashchev S.V.: Discrete spectrum for a periodic Schrödinger operator perturbed by a decreasing potential. Operator Theory: Adv. and Appl. 46, 109–114 (1990)

    MathSciNet  Google Scholar 

  13. Krein M.G.: Perturbation determinants and a formula for the traces of unitary and self-adjoint operators. Sov. Math. Dokl. 3, 707–710 (1962)

    Google Scholar 

  14. Krüger H., Teschl G.: Relative oscillation theory for Sturm–Liouville operators extended. J. Funct. Anal. 254-6, 1702–1720 (2008)

    Article  Google Scholar 

  15. Krüger, H., Teschl, G.: Effective Prüfer angles and relative oscillation criteria. J. Differ. Eqs. doi:10.1016/j.jde.2008.06.004, http://arxiv.org/abs/0709.0127v2[math.SP], 2007

  16. Leighton W.: On self-adjoint differential equations of second order. J. London Math. Soc. 27, 37–47 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  17. Reed M., Simon B.: Methods of Modern Mathematical Physics II. Fourier Analysis, Self-Adjointness. Academic Press, New York (1975)

    MATH  Google Scholar 

  18. Rofe-Beketov F.S.: A test for the finiteness of the number of discrete levels introduced into gaps of a continuous spectrum by perturbations of a periodic potential. Soviet Math. Dokl. 5, 689–692 (1964)

    MATH  Google Scholar 

  19. Rofe-Beketov F.S.: Spectral analysis of the Hill operator and its perturbations. FunkcionalÕnyï analiz 9, 144–155 (1977) (Russian)

    MATH  MathSciNet  Google Scholar 

  20. Rofe-Beketov, F.S.: A generalisation of the Prüfer transformation and the discrete spectrum in gaps of the continuous one. In: Spectral Theory of Operators, Baku: Elm, 1979 (Russian), pp.146–153

  21. Rofe-Beketov, F.S. Spectrum perturbations, the Kneser-type constants and the effective masses of zones-type potentials, Constructive Theory of Functions 84, (Sofia, 1984), Sofia: Verna, pp. 757–766

  22. Rofe-Beketov F.S.: Kneser constants and effective masses for band potentials. Sov. Phys. Dokl. 29, 391–393 (1984)

    MATH  ADS  Google Scholar 

  23. Rofe-Beketov F.S., Kholkin A.M.: Spectral analysis of differential operators. Interplay between spectral and oscillatory properties. Hackensack, World Scientific (2005)

    MATH  Google Scholar 

  24. Schmidt K.M.: Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm–Liouville operators. Commun. Math. Phys. 211, 465–485 (2000)

    Article  MATH  Google Scholar 

  25. Schmidt K.M.: Relative oscillation non-oscillation criteria for perturbed periodic Dirac systems. J. Math. Anal. Appl. 246, 591–607 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  26. Schmidt K.M.: An application of Gesztesy-Simon-Teschl oscillation theory to a problem in differential geometry. J. Math. Anal. Appl. 261, 61–71 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  27. Simon B.: Quantum Mechanics for Hamiltonians Defined as Quadratic Forms. Princeton University Press, Princeton, NJ (1971)

    MATH  Google Scholar 

  28. Simon B.: Trace Ideals and Their Applications. 2nd ed. Providence, RI, Amer. Math. Soc., (2005)

    MATH  Google Scholar 

  29. Simon, B.: Spectral Analysis of Rank One Perturbations and Applications. Lecture notes from Vancouver Summer School in Mathematical Physics, August 10–14, 1993, CRM Proc. 8, Providence, RI: Amer. Math. Soc., 1994

  30. Simon, B.: Sturm oscillation and comparison theorems. In: Sturm–Liouville Theory: Past and Present (eds. Amrein, W., Hinz, A., Pearson, D.), Basel: Birkhäuser, 2005, pp. 29–43

  31. Stolz G., Weidmann J.: Approximation of isolated eigenvalues of ordinary differential operators. J. Reine und Angew. Math. 445, 31–44 (1993)

    MATH  MathSciNet  Google Scholar 

  32. Sturm J.C.F.: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1, 106–186 (1836)

    Google Scholar 

  33. Teschl G.: Oscillation theory and renormalized oscillation theory for Jacobi operators. J. Diff. Eqs. 129, 532–558 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  34. Teschl G.: Renormalized oscillation theory for Dirac operators. Proc. Amer. Math. Soc. 126, 1685–1695 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  35. Teschl G.: Jacobi Operators and Completely Integrable Nonlinear Lattices Math. Surv. and Mon. 72. Providence, RI, Amer. Math. Soc., (2000)

    Google Scholar 

  36. Teschl G.: On the approximation of isolated eigenvalues of ordinary differential operators. Proc. Amer. Math. Soc. 136, 2473–2476 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  37. Teschl, G.: Relative oscillation theory for Dirac operators. In preparation

  38. Teschl, G.: Relative oscillation theory for Jacobi operators. In preparation.

  39. Walter W.: Ordinary Differential Equations. Springer, New York (1998)

    MATH  Google Scholar 

  40. Weidmann J.: Zur Spektraltheorie von Sturm–Liouville–Operatoren. Math. Z. 98, 268–302 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  41. Weidmann J.: Oszillationsmethoden für Systeme gewöhnlicher Differentialgleichungen. Math. Z. 119, 349–373 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  42. Weidmann J.: Spectral Theory of Ordinary Differential Operators. Lecture Notes in Mathematics, 1258. Springer, Berlin (1987)

    Google Scholar 

  43. Weidmann, J.: Spectral theory of Sturm–Liouville operators; approximation by regular problems. In: Sturm–Liouville Theory: Past and Present (eds. Amrein, W., Hinz, A., Pearson, D.), Basel: Birkhäuser, 2005, pp. 29–43

  44. Yafaev D.R.: Mathematical Scattering Theory: General Theory. Providence, RI, Amer. Math. Soc., (1992)

    MATH  Google Scholar 

  45. Zettl A.: Sturm–Liouville Theory. Providence, RI, Amer. Math. Soc., (2005)

    MATH  Google Scholar 

  46. Zheludev, V.A., Perturbation of the spectrum of the one-dimensional self-adjoint Schrödinger operator with a periodic potential. Topics in Mathematical Physics, Vol. 4, Birman, M.Sh. (ed), New York: Consultants Bureau, 1971, pp. 55–76

Download references

Author information

Author notes

  1. Helge Krüger

    Present address: Department of Mathematics, Rice University, Houston, TX, 77005, USA

Authors and Affiliations

  1. Faculty of Mathematics, Nordbergstrasse 15, 1090, Wien, Austria

    Helge Krüger & Gerald Teschl

  2. International Erwin Schrödinger Institute for Mathematical Physics, Boltzmanngasse 9, 1090, Wien, Austria

    Gerald Teschl

Authors
  1. Helge Krüger
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gerald Teschl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gerald Teschl.

Additional information

Communicated by B. Simon

Research supported by the Austrian Science Fund (FWF) under Grant No. Y330.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Krüger, H., Teschl, G. Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function. Commun. Math. Phys. 287, 613–640 (2009). https://doi.org/10.1007/s00220-008-0600-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0600-8

Keywords

Search

Navigation