Heat Equation on the Cone and the Spectrum of the Spherical Laplacian

TL;DR

This paper investigates the spectrum of the Laplacian on spherical domains using the heat equation on cones, introducing a spectral function that aids in estimating eigenvalues with high accuracy.

Contribution

It introduces a spectral function derived from the heat equation on cones, providing a new method for estimating Laplacian eigenvalues on spherical domains.

Findings

Spectral function can be expressed in closed form for certain domains.

The eigenvalue estimation method shows remarkable agreement with existing results.

Analytical properties suggest a simple scaling procedure for eigenvalue estimation.

Abstract

Spectrum of the Laplacian on spherical domains is analyzed from the point of view of the heat equation on the cone. The series solution to the heat equation on the cone is known to lead to a study of the Laplacian eigenvalue problem on domains on the sphere in higher dimensions. It is found that the solution leads naturally to a spectral function, a `generating function' for the eigenvalues and multiplicities of the Laplacian, expressible in closed form for certain domains on the sphere. Analytical properties of the spectral function suggest a simple scaling procedure for estimating the eigenvalues. Comparison of the first eigenvalue estimate with the available theoretical and numerical results for some specific domains shows remarkable agreement.