A remainder estimate for Weyl's law on Liouville tori

TL;DR

This paper improves the estimate of the error term in Weyl's law for eigenvalue distribution on Liouville tori, a broad class of integrable metrics, by reducing the problem to lattice point counting in planar domains.

Contribution

It introduces a novel approach combining separation of variables and advanced lattice point counting techniques to refine the remainder estimate in Weyl's law for Liouville tori.

Findings

Enhanced remainder estimate for Weyl's law on Liouville tori

Extension of classical flat metric results to broader Liouville metrics

Application of lattice point counting methods to spectral geometry

Abstract

The paper is concerned with the asymptotic distribution of Laplace eigenvalues on Liouville tori. Liouville metrics are the largest known class of integrable metrics on two-dimensional tori; they contain flat metrics and metrics of revolution as special cases. Using separation of variables, we reduce the eigenvalue counting problem to the problem of counting lattice points in certain planar domains. This allows us to improve the remainder estimate in Weyl's law on a large class of Liouville tori. For flat metrics, such an estimate has been known for more than a century due to classical results of W. Sierpinski and J.G. van der Corput. Our proof combines the method of Y. Colin de Verdiere, who proved an analogous result for metrics of revolution on a sphere, with the techniques developed by P. Bleher, D. Kosygin, A. Minasov and Y. Sinai in their study of the almost periodic properties of the remainder in Weyl's law on Liouville tori.