A lower bound for sums of eigenvalues of the Laplacian

Abstract

Let $\lambda_k(\Omega)$ be the $k$th Dirichlet eigenvalue of a bounded domain $\Omega$ in $\mathbb{R}^n$. According to Weyl's asymptotic formula we have $$ \lambda_k(\Omega)\sim C_n(k/V(\Omega))^{2/n}. $$ The optimal in view of this asymptotic relation lower estimate for the sums $\sum_{j=1}^k \lambda_j(\Omega)$ has been proven by P.Li and S.T.Yau (Comm. math. Phys. 88 *1983), 309-318). Here we will improve this estimate by addint to its right-hand side a term of the order of k that depends on the ratio of the volume to the moment of inertia of $\Omega$.