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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.

A remainder estimate for Weyl’s law on Liouville tori