Asymptotics for the spectral function on Zoll manifolds

TL;DR

This paper investigates the asymptotic behavior of spectral functions and projectors on Zoll manifolds, revealing how eigenvalue counts and kernels behave in the high-frequency limit, with comparisons to standard geometries.

Contribution

It establishes new asymptotic formulas for eigenvalue counting functions and spectral projectors on Zoll manifolds, extending classical results to more general geometric settings.

Findings

Eigenvalue counting functions follow a specific asymptotic pattern for large eigenvalues.

Spectral projectors exhibit sphere- and torus-like asymptotics near points with few loops.

A sum over multiple eigenvalue windows yields the same leading order as classical cases.

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.