Asymptotics for the spectral function on Zoll manifolds

Abstract

On a smooth, compact, Riemannian manifold without boundary $(M,g)$, let ${\Delta }_g$ be the Laplace–Beltrami operator. We define the orthogonal projection operator

for an interval $I_{\lambda }$ centered around $\lambda \in \mathbb{R}$ of a small, fixed length. The Schwartz kernel, ${\Pi }_{I_{\lambda }}(x,y)$, of this operator plays a key role in the analysis of monochromatic random waves, a model for high energy eigenfunctions. It is expected that ${\Pi }_{I_{\lambda }}(x,y)$ has universal asymptotics as $\lambda \to \infty$ in a shrinking neighborhood of the diagonal in $M\times M$ (provided $I_{\lambda }$ is chosen appropriately) and hence that certain statistics for monochromatic random waves have universal behavior. These asymptotics are well known for the torus and the round sphere, and were recently proved to hold near points in $M$ with few geodesic loops by Canzani–Hanin. In this article, we prove that the same universal asymptotics hold in the opposite case of Zoll manifolds (manifolds all of whose geodesics are closed with a common period) under an assumption on the volume of loops with length incommensurable with the minimal common period.