Geodesic
Convergence of regularized traces of powers of the Laplace–Beltrami operator with potential on the sphere $S^n$
Sbornik. Mathematics, Tome 190 (1999) no. 10, pp. 1401-1415 
A. N. Bobrov; V. E. Podolskii. Convergence of regularized traces of powers of the Laplace–Beltrami operator with potential on the sphere $S^n$. Sbornik. Mathematics, Tome 190 (1999) no. 10, pp. 1401-1415. http://geodesic.mathdoc.fr/item/SM_1999_190_10_a0/
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     title = {Convergence of regularized traces of powers of {the~Laplace{\textendash}Beltrami} operator with potential on the sphere~$S^n$},
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For the Laplace–Beltrami operator $-\Delta$ on the sphere $S^n$ perturbed by the operator of multiplication by an infinitely smooth complex-valued function $q$, the convergence without brackets of regularized traces $$ \sum_k\biggl(\mu_k^\alpha -\lambda_k^\alpha-\sum_j\chi_j(\alpha )\lambda_k^{k_j(\alpha)}\biggr), $$ is studied, where the $\mu_k$ and the $\lambda_k$ are the eigenvalues of the operators $-\Delta+q$ and $-\Delta$, respectively. Sharp estimates of $\alpha$ in the cases of absolute and conditional convergence are obtained. Explicit formulae for the coefficients $\chi_j$ are obtained for odd potentials $q$.

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