Band asymptotics of eigenvalues for the Zoll manifold (Q1118881)

Let \(\Delta\) be the Laplace-Beltrami operator. Let \((S^ n,g)\) be a Zoll metric i.e. all the geodesics are closed with length \(2\pi\). Since the geodesic geometry closely resembles the standard sphere, it is natural to imagine that the same is true of the spectral geometry. Let \(\lambda_ k=k(k+n-1)\) for \(k\geq 0\) be the eigenvalues of the Laplacian for the canonical metric on the sphere. Weinstein showed \(\exists M>0\) so the eigenvalues of \(\Delta\) belong to the intervals \([\lambda_ k- M,\lambda_ k+M];\) let \(\{\mu_{k,j}\}\) be these eigenvalues. Work by Colin de Verdière and Kubawara gives asymptotic information regarding the arithmetic means of the \(\{\mu_{k,j}\}\). In the present paper, the author obtains more precise information concerning the asymptotic distribution of the \(\mu_{k,j}\) and studies the asymptotic band structure of these clusters.