On sums of eigenvalues of elliptic operators on manifolds

TL;DR

This paper derives upper bounds for sums of eigenvalues of various elliptic operators on manifolds, extending known results and providing asymptotically sharp estimates aligned with Weyl law predictions.

Contribution

It introduces a unified approach using the averaged variational principle to obtain sharp eigenvalue bounds for multiple operators on manifolds, including Laplace-Beltrami, Witten Laplacian, and Schrödinger operators.

Findings

Derived upper bounds for sums of eigenvalues matching Weyl law asymptotics.

Extended bounds to operators on conformal manifolds and infinite domains.

Provided sharp estimates for heat kernel traces as corollaries.

Abstract

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kr{\"o}ger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schr{\"o}dinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.