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Potential Analysis

, Volume 39, Issue 2, pp 195–202 | Cite as

An Asymptotic Formula Concerning the Eigenvalues of the Dirichlet Laplacian in a Planar Domain

  • M. R. Dostanić
Article

Abstract

We shall prove the equivalent of the Carleman-Grinberg formula for the Dirichlet Laplacian in convex planar domain.

Keywords

Dirichlet Laplacian Green function McDonald function Nuclear operators 

AMS Subject Classification

47A75 35J25 

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References

  1. 1.
    Berezin, F.A.: Convex function of operators. Mat. Sb. (N.S.) 88(130), 268–476 (1972)MathSciNetGoogle Scholar
  2. 2.
    Berezin, F.A.: Covariant and contravariant symbols of operators. Izv. Akad.Nauki. SSSR, Ser. Mat. 36, 1134–1167 (1972)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Carleman, T.: Veber die asymptotische Verteilung der Eigenwerte partieller Differentialgechungen, Berichte der Sächsisch. Akad. der wiss. zu Lepzig, Math. Phys, vol. 119. Klasse LXXXVIII (1936)Google Scholar
  4. 4.
    Dostanić, M.R.: Regularized trace of the inverse of the Dirichlet Laplacian. Commun. Pure and Appl. Math. 64(8), 1148–1164 (2011)zbMATHCrossRefGoogle Scholar
  5. 5.
    Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear nonselfadjoint operators. Transl. of Math. Monographs, 19, Amer. Math. Soc. Providence R. I. (1969)Google Scholar
  6. 6.
    Grinberg, S.I.: Asymptotic of the eigenvalues of the Laplace operator. Uspekhi Mat Nauk, T VIII, V. 6, 97–102 (1953)MathSciNetGoogle Scholar
  7. 7.
    Krein, M.G.: On the trace formula in perturbation theory. Mat. Sb. N.S. 33(75), 597–626 (1953)MathSciNetGoogle Scholar
  8. 8.
    Randol, B.: On the Fourier transform of the indicator function of a planar set. Trans. Am. Math. Soc. 139, 271–278 (1969)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Simon, B.: Trace ideals and their applications, 2nd edn. In: Mathematical Surveys and Monographs, vol. 120, Amer. Math. Soc., Providence R.I. (2005)Google Scholar
  10. 10.
    Gesztesy, F., Pushnitsky, A., Simon, B.: On the Koplienka spectral shift function, I. Basics. Zh. Mat. Fiz. Anal. Geom. 4(1), 63–107 (2008)zbMATHGoogle Scholar
  11. 11.
    Watson, G.N.: A treatise of the theory of Bessel functions, 2nd edn. Cambridge University Press, Cambridge (1962)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Matematicki fakultetUniverzitet u BeograduBeogradSerbia

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