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Journal of Spectral Theory

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Volume 8, Issue 4, 2018, pp. 1529–1550
DOI: 10.4171/JST/234

Published online: 2018-10-22

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

Evans M. Harrell II[1] and Joachim Stubbe[2]

(1) Georgia Institute of Technology, Atlanta, USA
(2) EPFL, Lausanne, Switzerland

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.

Keywords: Neumann Laplacian, Dirichlet Laplcian, semiclassical bounds for eigenvalues

Harrell II Evans, Stubbe Joachim: Two-term, asymptotically sharp estimates for eigenvalue means of the Laplacian. J. Spectr. Theory 8 (2018), 1529-1550. doi: 10.4171/JST/234