Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian

TL;DR

This paper derives asymptotically sharp bounds for the eigenvalues of the Laplacian with Neumann boundary conditions, improving semiclassical estimates and providing two-sided bounds for individual eigenvalues.

Contribution

It introduces new asymptotically sharp inequalities for Laplacian eigenvalues using the averaged variational principle, enhancing existing semiclassical bounds.

Findings

Improved semiclassical bounds for the Riesz mean of eigenvalues

Two-sided bounds for individual eigenvalues that are semiclassically sharp

Remarks on the Dirichlet case using similar methods

Abstract

We present asymptotically sharp inequalities for the eigenvalues $\mu_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \cite{HaSt14}. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$. In addition, we obtain two-sided bounds for individual $\mu_k$, which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.