Regularized traces of a perturbed Laplace-Beltrami operator on the unit two-dimensional sphere (Q5942075)

Let \(T\) be the Laplace-Beltrami operator on the unit sphere \(S^2\) in Euclidean space. The eigenvalues \(\lambda_k=k(k+1)\) of \(T\) have multiplicity \(\nu_k=2k+1\). Let \(P\) be the perturbation defined by the operator of multiplication by a complex function \(p\in L^2(S^2)\). The operator \(T+P\) has a discrete spectral resolution. Let \(\mu_{k,i}\) be the eigenvalues of \(T+P\) with algebraic multiplicity \(\nu_k\) so \(|\mu_{k,i}-\lambda_k|\leq const\) for \(i=0,1,\dots ,\nu_k-1\). The authors establish the following clustering formula for potentials which have a suitable equivariance property: Theorem. Assume there exists a rotation \(K\) of \(S^2\) so \(Kp(x)=p(Kx)=tp(x)\) for \(|t|=1\) and \(t^2\neq 1\). Then \[ \lim_{n\rightarrow\infty}\sum_{k=0}^n\Biggl\{ \sum_{i=0}^{2k}\mu_{k,i}-k(k+1)(2k+1)\Biggr\}=0. \]