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

Authors:	V A Sadovnichii Z Yu Fazullin

Institution:	(1) M. V. Lomonosov Moscow State University, Moscow;(2) Bashkir State University, Ufa

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.Translated from Matematicheskie Zametki, vol. 77, no. 3, 2005, pp. 434–448Original Russian Text Copyright © 2005 by V. A. Sadovnichii, Z. Yu. Fazullin.This revised version was published online in April 2005 with a corrected issue number.

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