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Mathematical Notes

, Volume 77, Issue 3–4, pp 400–413 | Cite as

Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere \(\mathbb{S}^2 \)

  • V. A. Sadovnichii
  • Z. Yu. Fazullin
Article

Abstract

In this paper, we study the asymptotics of the eigenvalues of the Laplace operator perturbed by an arbitrary bounded operator on the sphere \(\mathbb{S}^2 \). For the first time, for the partial differential operator of second order, the leading term of the second correction of perturbation theory is obtained. A connection between the coefficient of the second term of the asymptotics of the eigenvalues and the formula for the traces of the operator under consideration is established.

Key words

Laplace operator perturbation theory trace of perturbed Laplace operators eigenvalues of the Laplace operator Legendre polynomials 

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Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • V. A. Sadovnichii
    • 1
  • Z. Yu. Fazullin
    • 2
  1. 1.M. V. Lomonosov Moscow State UniversityMoscow
  2. 2.Bashkir State UniversityUfa

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