Convergence of regularized traces of powers of the Laplace–Beltrami operator with potential on the sphere $S^n$

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 $q$.