Klein-Gordon算子Berezin-Li-Yau型不等式的改进
An Improvement to a Berezin-Li-Yau type inequality for the Klein-Gordon Operator
摘要：
本文改进了[E. M. Harrell II和S.杨文，张文，张文，等。分析，256（12）（2009）3977-3995]。这里，$β_j$表示当被限制到有界集合$Ω\subset {\mathbb R}^n$时克莱因戈登哈密顿算子$H_{0,Ω}=|p|$的第$j$个特征值。$H_{0,Ω}$也可以被描述为柯西随机过程的生成器，在$\partial Ω$上具有杀死条件。（参见：[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，柯西过程和Steklov问题，J. Funct。Anal. 234（2006）199-225]。为此，我们采用Melas（[ A]）的证明。D. Melas，A lower bound for sums of eigenvalues of the Laplacian，Proceedings of the American Mathematical Society，131（2）（2002）631-636]），他改进了对$\sum_{j=1}^kλ_j$的界的估计，其中$λ_j$表示在${\mathbb R}^d$中的有界域上的Dirichlet Laplacian的$j$第i个特征值。
英文摘要：
In this article we improve a lower bound for $\sum_{j=1}^kβ_j$ (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 $β_j$ denotes the $j$th eigenvalue of the Klein Gordon Hamiltonian $H_{0,Ω}=|p|$ when restricted to a bounded set $Ω\subset {\mathbb R}^n$. $H_{0,Ω}$ can also be described as the generator of the Cauchy stochastic process with a killing condition on $\partial Ω$. (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 $\sum_{j=1}^kλ_j$, where $λ_j$ denotes the $j$th eigenvalue of the Dirichlet Laplacian on a bounded domain in ${\mathbb R}^d$.
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