Mathematical Physics, Analysis and Geometry

, Volume 10, Issue 3, pp 261–270 | Cite as

Estimating Eigenvalue Moments via Schatten Norm Bounds on Semigroup Differences

  • M. Hansmann


We derive new bounds on the moments of the negative eigenvalues of a selfadjoint operator B. The moments of order \(0<\gamma \leqslant 1\) are estimated in terms of Schatten-norm bounds on the difference of the semigroups generated by B and a reference operator A which is assumed to be nonnegative and selfadjoint. The estimate in the case γ = 1 is sharp.


Eigenvalues Moments Schatten ideals Selfadjoint operators Semigroup difference Spectral shift function 

Mathematics Subject Classifications (2000)

47A10 47A75 81Q10 


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  1. 1.
    Birman, M.Sh., Solomyak, M.Z.: Remarks on the spectral shift function. J. Sov. Math. 3, 408–419 (1975)MATHCrossRefGoogle Scholar
  2. 2.
    Birman, M.Sh., Solomyak, M.Z.: Double operator integrals in a Hilbert space. Integral Equations Operator Theory 47, 131–168 (2003)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Birman, M.Sh., Yafaev, D.: The spectral shift function. The papers of M.G. Krein and their further development. (Russian) Algebra i Analiz 4(5), 1–44 (1992)MathSciNetGoogle Scholar
  4. 4.
    Combes, J.M., Hislop, P.D., Nakamura, S.: The \(L\sp p\)-theory of the spectral shift function, the Wegner estimate, and the integrated density of states for some random operators. Comm. Math. Phys. 218(1), 113–130 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Demuth, M., Katriel, G.: Eigenvalue Inequalities in Terms of Schatten Norm Bounds on Differences of Semigroups, and Application to Schrödinger Operators. To appear in Annales Henri Poincare.
  6. 6.
    Demuth, M., van Casteren, J.: Stochastic Spectral Theory for Selfadjoint Feller Operators: A Functional Integration Approach. Birkhäuser Verlag, Basel (2000)MATHGoogle Scholar
  7. 7.
    Hundertmark, D., Killip, R., Nakamura, S., Stollmann, P., Veselić, I.: Bounds on the spectral shift function and the density of states. Comm. Math. Phys. 262(2), 489–503 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  8. 8.
    Yafaev, D.: Mathematical Scattering Theory. General Theory. Translations of Mathematical Monographs, vol. 105. American Mathematical Society, Providence, RI (1992)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Institute of MathematicsTechnical University of ClausthalClausthal-ZellerfeldGermany

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