An improvement to a Berezin-Li-Yau type inequality

Abstract:

In this article we improve a lower bound for $\sum _{j=1}^k\beta _j$ (a Berezin-Li-Yau type inequality) that appeared in an earlier paper of Harrell and Yolcu. Here $\beta _j$ denotes the $j$th eigenvalue of the Klein Gordon Hamiltonian $H_{0,\Omega }=|p|$ when restricted to a bounded set $\Omega \subset {\mathbb R}^n$. $H_{0,\Omega }$ can also be described as the generator of the Cauchy stochastic process with a killing condition on $\partial \Omega$. To do this, we adapt the proof of Melas, who improved the estimate for the bound of $\sum _{j=1}^k\lambda _j$, where $\lambda _j$ denotes the $j$th eigenvalue of the Dirichlet Laplacian on a bounded domain in ${\mathbb R}^d$.