Asymptotics for the spectral function on Zoll manifolds

Abstract

Let $(M,g)$ be a Zoll manifold, i.e., a smooth, compact, Riemannian manifold without boundary all of whose geodesics are closed with a minimal common period $T$. The positive definite Laplace-Beltrami operator has eigenvalues $\{\lambda_j^2\}_j$ which cluster around $\nu^2_\ell$ for some sequence $\nu_\ell\to \infty$. This article is concerned with the number of $\lambda_j$ in a window of fixed size $\mathrm{w}$ around $\nu_\ell$, denoted by $\mathbf{N}(\nu_\ell,\mathrm{w}):=\#\{j\,:\, \lambda_j\in[\nu_\ell-\mathrm{w},\nu_\ell+\mathrm{w}]\}.$ When the set of trajectories with period smaller than $T$ has zero measure, there is $c_{n}>0$, depending only on $n=\operatorname{dim} M$, such that $$ \mathbf{N}(\nu_\ell,\mathrm{w}) =c_n\operatorname{vol}_g(M)\nu_{\ell}^{n-1}+o(\nu_{\ell}^{n-1}), $$ as $\ell \to \infty$. However, for a general Zoll manifold this may not be the case. We show that, nevertheless, there is $N>0$, independent of $\ell$, such that $$ \sum_{j=0}^{N-1}\mathbf{N}(\nu_{\ell+j},\mathrm{w})= c_nN\operatorname{vol}_g(M)\nu_{\ell}^{n-1}+o(\nu_{\ell}^{n-1}), $$ as $\ell \to \infty$. In addition to asymptotics for the counting function, we study the kernel of the spectral projector for the Laplacian, $\Pi_{\ell,\mathrm{w}}(x,y)$ onto the spectrum in ${\bigcup_{j=0}^{N-1}[\nu_{\ell+j}-\mathrm{w},\nu_{\ell+j}+\mathrm{w}]}$. We show that for $x$ and $y$ in a shrinking neighborhood of a point with few loops of length smaller than $T$, $\Pi_{\ell,\mathrm{w}}(x,y)$ and its derivatives have the same asymptotics as those on the round sphere and flat torus.