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

TL;DR

This paper constructs infinite-volume domains with discrete Laplacian spectra and derives spectral asymptotics, extending Weyl's formula to new geometric settings using elementary techniques.

Contribution

It introduces a method to create infinite-volume domains with discrete spectra and provides precise spectral asymptotics, generalizing Weyl's law.

Findings

Constructed domains with purely discrete Laplacian spectrum

Derived spectral asymptotics in terms of domain geometry

Extended Weyl's law to infinite-volume settings

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.