Semiclassical Estimates for Eigenvalue Means of Laplacians on Spheres

We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of S d . We also prove a Berezin–Li–Yau inequality for domains contained in the hemisphere S + 2 .