Eigenvalue clusters for the hemisphere Laplacian with variable Robin condition
Alexander Pushnitski, Igor Wigman
TL;DR
This work analyzes the Robin Laplacian on the hemisphere with a variable Robin coefficient on the equator and shows that eigenvalues collapse into clusters of size $\ell+1$ around the Neumann spectrum $\ell(\ell+1)$. The authors reduce the Robin–Neumann gaps to the spectrum of a finite-dimensional operator $V_\ell[\sigma]$ acting on the Neumann $\ell$-eigenspace, then recast this operator as a semiclassical pseudodifferential operator on the equator with semiclassical parameter $1/\ell$ and apply trace asymptotics to obtain an explicit density for the eigenvalues inside each cluster. The density depends only on the even part of the Robin coefficient, $\sigma_{\mathrm{even}}$, via a precise two-variable integral, and the paper also treats the special case of odd $\sigma$, showing that for odd trigonometric polynomials only $d+1$ gaps can be nonzero. A careful discussion contrasts the results with Weinstein’s formula and highlights subtle limit-interchange phenomena by providing a one-dimensional analogy. Overall, the work advances semiclassical methods in spectral geometry by characterizing high-energy eigenvalue distributions under a spatially varying boundary condition.
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.