An Improvement to a Berezin-Li-Yau type inequality for the Klein-Gordon Operator

In this article we improve a lower bound for $\sum_{j=1}^k\beta_j$ (a Berezin-Li-Yau type inequality) in [E. M. Harrell II and S. Yildirim Yolcu, Eigenvalue inequalities for Klein-Gordon Operators, J. Funct. Analysis, 256(12) (2009) 3977-3995]. 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 $\partial \Omega$. (cf. [R. Banuelos, T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincare inequality, J. Funct. Anal. 211 (2) (2004) 355-423]; [R. Banuelos, T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Anal. 234 (2006) 199-225].) To do this, we adapt the proof of Melas ([ A. D. Melas, A lower bound for sums of eigenvalues of the Laplacian, Proceedings of the American Mathematical Society, 131(2) (2002) 631-636]), 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 ${\mathbb R}^d$.