Convergence of regularized traces of powers of the Laplace-Beltrami operator with potential on the sphere Sn

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 [GRAPHICS] 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.