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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

   

 

An improvement to a Berezin-Li-Yau type inequality


Author: Selma Yildirim Yolcu
Journal: Proc. Amer. Math. Soc. 138 (2010), 4059-4066
MSC (2010): Primary 35P15; Secondary 35S99
Published electronically: May 18, 2010
MathSciNet review: 2679626
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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}=\vert p\vert$ 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$.


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Additional Information

Selma Yildirim Yolcu
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Address at time of publication: Department of Mathematics, Georgia College & State University, Milledgeville, Georgia 31061; (after August 2010) Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: selma@math.gatech.edu, selma.yildirim-yolcu@gcsu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10419-7
Keywords: Fractional Laplacian, Weyl law, universal bounds, Klein-Gordon operator, Berezin-Li-Yau inequality
Received by editor(s): September 19, 2009
Received by editor(s) in revised form: January 24, 2010
Published electronically: May 18, 2010
Dedicated: This paper is dedicated to Professor Evans M. Harrell
Communicated by: Matthew J. Gursky
Article copyright: © Copyright 2010 American Mathematical Society