We study the eigenvalue clusters of the Robin Laplacian on the two-dimensional
hemisphere with a variable Robin parameter on the equator. The
-th cluster
has
eigenvalues. We determine the asymptotic density of eigenvalues in the
-th cluster
as
tends to infinity. This density is given by an explicit integral involving the even part
of the Robin parameter.
Keywords
Robin boundary conditions, Robin–Neumann gaps, hemisphere,
spherical harmonics, eigenvalue clusters, variable Robin
parameter
