A. N. Bobrov, V. E. Podolskii, “Convergence of regularized traces of powers of the Laplace–Beltrami operator with potential on the sphere <nobr>$S^n$</nobr>”, Mat. Sb., 1999, Volume 190, Number 10,Pages <nobr>3

This article is cited in 4 papers

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

A. N. Bobrov, V. E. Podolskii

M. V. Lomonosov Moscow State University

Abstract: 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 $q$.

UDC: 517.956.227

MSC: 58G25, 58G03, 35P20

Received: 07.05.1998

DOI: 10.4213/sm430