Geodesic
Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere~$\mathbb S^2$
Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 434-448.

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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. 
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V. A. Sadovnichii; Z. Yu. Fazullin. Asymptotics of the eigenvalues and the formula for the trace of perturbations of the Laplace operator on the sphere~$\mathbb S^2$. Matematičeskie zametki, Tome 77 (2005) no. 3, pp. 434-448. http://geodesic.mathdoc.fr/item/MZM_2005_77_3_a9/

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