Semiclassical estimates for eigenvalue means of Laplacians on spheres

TL;DR

This paper derives precise semiclassical asymptotic formulas and bounds for eigenvalues of Laplacians on spheres and related domains, extending classical inequalities and analyzing polyharmonic operators.

Contribution

It provides new three-term asymptotic expansions, sharp bounds, and inequalities for Laplacian eigenvalues on spheres, hemispheres, and other symmetric spaces, including polyharmonic operators.

Findings

Established three-term semiclassical asymptotics for eigenvalue counting functions.

Proved sharp bounds and inequalities for eigenvalues on spheres and hemispheres.

Extended results to polyharmonic operators and sum rules for eigenvalues.

Abstract

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 $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for P\'olya-type inequalities. Finally, we provide sum rules for Laplacian eigenvalues on spheres and compact two-point homogeneous spaces.