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

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 [14]. 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, and we obtain a Neumann version of Laptev’s result that the Pólya conjecture is valid for domains that are Cartesian products of a generic domain with one for which Pólya’s conjecture holds. In a final section, we remark upon the Dirichlet case with the same methods.