Eigenvalue clusters for the hemisphere Laplacian with variable Robin condition

TL;DR

This paper analyzes the eigenvalue clusters of the Robin Laplacian on a hemisphere with a variable Robin boundary condition, deriving the asymptotic density of eigenvalues in each cluster as the cluster index grows large.

Contribution

It provides an explicit asymptotic formula for the eigenvalue density in clusters for the hemisphere Robin Laplacian with variable boundary conditions, advancing spectral analysis techniques.

Findings

Eigenvalue clusters contain +1 eigenvalues each.

Asymptotic density of eigenvalues is given by an explicit integral.

Density depends on the even part of the Robin coefficient.

Abstract

We study the eigenvalue clusters of the Robin Laplacian on the 2-dimensional hemisphere with a variable Robin coefficient on the equator. The $\ell$'th cluster has $\ell+1$ eigenvalues. We determine the asymptotic density of eigenvalues in the $\ell$'th cluster as $\ell$ tends to infinity. This density is given by an explicit integral involving the even part of the Robin coefficient.