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

2016/07/08 by Harrell, Evans M., Stubbe, Joachim
#47A75 #47F05 #58J50 #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1607.02207

Abstract

We present asymptotically sharp inequalities for the eigenvalues μ_k of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \citeHaSt14. For the Riesz mean R₁(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 μ_k, which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.

Related

• Complementary asymptotically sharp estimates for eigenvalue means of Laplacians
• Semiclassical estimates for eigenvalue means of Laplacians on spheres
• Semiclassical inequalities for Dirichlet and Neumann Laplacians on convex domains
• Weil asymptotic formula for the Laplacian on domains with rough boundaries
• Riesz means asymptotics for Dirichlet and Neumann Laplacians on Lipschitz domains