We can hear (some of) the shape of dented horns

Abstract

In this article we construct a family of domains $\Omega \subset \mathbb{R}^2$ with infinite volume such that the Dirichlet Laplacian $\Delta^D$ has purely discrete spectrum and give precise spectral asymptotics for the eigenvalue counting function in terms of the geometry of $\Omega$. This generalizes the well-known asymptotic formula of Hermann Weyl to this class of infinite volume domains. The construction is elementary, uses only the bracketing technique invented by Weyl himself and it is extendable to arbitrary dimensions.