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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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An improvement to a Berezin-Li-Yau type inequality
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by Selma Yıldırım Yolcu PDF
Proc. Amer. Math. Soc. 138 (2010), 4059-4066 Request permission

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 $H_{0,\Omega }=|p|$ 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 $\partial \Omega$. 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 Yıldırım 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
  • 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
  • © Copyright 2010 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 138 (2010), 4059-4066
  • MSC (2010): Primary 35P15; Secondary 35S99
  • DOI: https://doi.org/10.1090/S0002-9939-2010-10419-7
  • MathSciNet review: 2679626