Families of non-tiling domains satisfying P\'olya's conjecture

Abstract

We show the existence of classes of non-tiling domains satisfying P\'{o}lya's conjecture in any dimension, in both the Euclidean and non-Euclidean cases. This is a consequence of a more general observation asserting that if a domain satisfies P\'{o}lya's conjecture eventually, that is, for a sufficiently large order of the eigenvalues, and may be partitioned into $p$ non-overlapping isometric sub-domains, with $p$ arbitrarily large, then there exists an order $p_{0}$ such that for $p$ larger than $p_{0}$ all such sub-domains satisfy P\'{o}lya's conjecture. In particular, this allows us to show that families of sectors of domains of revolution with analytic boundary, and thin cylinders satisfy P\'{o}lya's conjecture, for instance. We also improve upon the Li-Yau constant for general cylinders in the Dirichlet case.