Weyl formula improvement for product of Zoll manifolds

TL;DR

This paper extends the improvement of the Weyl law remainder term from product of round spheres to general Zoll manifolds by analyzing eigenvalue clusters and lattice point distributions.

Contribution

It generalizes previous results on Weyl law improvements to Zoll manifolds, broadening the class of manifolds with refined spectral asymptotics.

Findings

Improved Weyl law remainder for Zoll manifolds

Reduction of spectral problem to lattice point distribution

Extension of polynomial improvement to Zoll manifolds

Abstract

Iosevich and Wyman have proved in ~\cite{IoWy} that the remainder term in classical Weyl law can be improved from $O(\lambda^{d-1})$ to $o(\lambda^{d-1})$ in the case of product manifold by using a famous result of Duistermaat and Guillemin. They also showed that we could have polynomial improvement in the special case of Cartesian product of round spheres by reducing the problem to the study of the distribution of weighted integer lattice points. In this paper, we show that we can extend this result to the case of Cartesian product of Zoll manifolds by investigating the eigenvalue clusters of Zoll manifold and reducing the problem to the study of the distribution of weighted integer lattice points too.