An improvement to a Berezin-Li-Yau type inequality
Selma Yıldırım Yolcu · Proceedings of the American Mathematical Society · 2010
In this article we improve a lower bound for ∑ j = 1 k β j \sum _{j=1}^k\beta _j (a Berezin-Li-Yau type inequality) that appeared in an earlier paper of Harrell and Yolcu. Here β j \beta _j denotes the j j th eigenvalue of the Klein Gordon Hamiltonian H 0 , Ω = | p | H_{0,\Omega }=|p| when restricted to a bounded set Ω ⊂ R n \Omega \subset {\mathbb R}^n . H 0 , Ω H_{0,\Omega } can also be described as the generator of the Cauchy stochastic process with a killing condition on ∂ Ω \partial \Omega . To do this, we adapt the proof of Melas, who improved the estimate for the bound of ∑ j = 1 k λ j \sum _{j=1}^k\lambda _j , where λ j \lambda _j denotes the