AN IMPROVEMENT TO A BEREZIN-LI-YAU TYPE INEQUALITY

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