Spectral asymptotics of the Dirichlet Laplacian in a conical layer

TL;DR

This paper investigates the spectral properties of the Dirichlet Laplacian in conical layers, revealing eigenvalue accumulation near the essential spectrum and deriving asymptotics for the first eigenvalues in the small aperture limit.

Contribution

It provides new insights into eigenvalue accumulation and asymptotic behavior of eigenfunctions in conical layers, including precise localization estimates and asymptotic formulas.

Findings

Eigenvalues accumulate below the essential spectrum threshold.

Number of eigenvalues near the threshold grows logarithmically with distance.

Eigenfunctions localize in the conical cap at a scale related to the aperture angle.

Abstract

The spectrum of the Dirichlet Laplacian on conical layers is analysed through two aspects: the infiniteness of the discrete eigenvalues and their expansions in the small aperture limit. On the one hand, we prove that, for any aperture, the eigenvalues accumulate below the thresh-old of the essential spectrum: For a small distance from the essential spectrum, the number of eigenvalues farther from the threshold than this distance behaves like the logarithm of the distance. On the other hand, in the small aperture regime, we provide a two-term asymptotics of the first eigenvalues thanks to a priori localization estimates for the associated eigenfunctions. We prove that these eigenfunctions are localized in the conical cap at a scale of order the cubic root of the aperture angle and that they get into the other part of the layer at a scale involving the logarithm of the aperture angle.