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

2014/05/29 by Nils Rautenberg, Rautenberg, Nils
Biochemistry, Genetics and Molecular Biology · Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Modeling in Engineering #Diffusion and Search Dynamics #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Point processes and geometric inequalities #Spectral Theory (math.SP) #math-ph #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1405.7531

Update, minor corrections and typos

openalex publication_date 2014/05/29 · arxiv created 2014/05/31 · arxiv updated 2014/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this article we construct a family of domains Ω⊂ ℝ² with infinite volume such that the Dirichlet Laplacian Δ^D has purely discrete spectrum and give precise spectral asymptotics for the eigenvalue counting function in terms of the geometry of Ω. 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.

Citations

• Can One Hear the Shape of a Drum?

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