Some spectral results for the Laplacian on line bundles over \(S^ n\) (Q799957)

Let E be a Hermitian line bundle over the standard sphere \(S^ n\), and let \(\tilde d\) be a linear connection compatible with the Hermitian structure. Then one can naturally define a non-negative, second order, self-adjoint, elliptic differential operator L called the Laplacian. When E is trivial and \(\tilde d\) is a flat connection, L is just the Laplace- Beltrami operator whose spectrum consists of the eigenvalues \(\lambda_ k^{(0)}=k(k+n-1) (k=0,1,2,...)\). The line bundle over \(S^ n\) is always trivial when \(n\neq 2\), and the set of equivalence classes of line bundles over \(S^ 2\) is \(\{E_ m| m\in {\mathbb{Z}}\}\). On each line bundle \(E_ m\) there is a unique harmonic connection \(\tilde d{}_ m\), and the eigenvalues of the associated Laplacian are calculated as \(\lambda_ k^{(m)}=\{k+(| m| +1)/2\}^ 2-(m^ 2+1)/4(k=0,1,2,...).\) Let \(\mu_ j^{(k)} (j=1,...,N_ k^{(m)})\) be the eigenvalues of L on \(E_ m\) which is split from \(\lambda_ k^{(m)}\). Let \(Q_{\tilde d}(\gamma) (\in S^ 1)\) denote the holonomy of the connection \(\tilde d\) along a closed curve \(\gamma\) of \(S^ n\). The first result of the paper is that \((1)\quad\max_{j}|\mu_ j^{(k)}-\lambda_ k^{(m)}| =O(k)\) (\(k\to\infty )\), and \((2)\quad\max_{j}|\mu_ j^{(k)}-\lambda_ k^{(m)}| =O(1)\) holds if and only if \(Q_{\tilde d}(\gamma)=Q_{\tilde d_ m}(\gamma) (=(-1)^ m)\) holds for every closed geodesic \(\gamma\) of \(S^ n\). Furthermore when the case (2) occurs the paper clarifies the asymptotic distribution of \(\{\mu_ j^{(k)}\}\) around \(\lambda_ k^{(m)}\) as \(k\to\infty \).