Eigenvalue clusters for the hemisphere Laplacian with variable Robin condition

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