An improvement to a BerezinLiYau type inequality
Author:
Selma Yildirim Yolcu
Journal:
Proc. Amer. Math. Soc. 138 (2010), 40594066
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 (a BerezinLiYau type inequality) that appeared in an earlier paper of Harrell and Yolcu. Here denotes the th eigenvalue of the Klein Gordon Hamiltonian when restricted to a bounded set . can also be described as the generator of the Cauchy stochastic process with a killing condition on . To do this, we adapt the proof of Melas, who improved the estimate for the bound of , where denotes the th eigenvalue of the Dirichlet Laplacian on a bounded domain in .
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 2.
 R. Bañuelos and T. Kulczycki, The Cauchy process and the Steklov problem, J. Funct. Analysis 211(2) (2004) 355423. MR 2056835 (2005b:60124)
 3.
 R. Bañuelos and T. Kulczycki, Eigenvalue gaps for the Cauchy process and a Poincaré inequality, J. Funct. Analysis 234(1) (2006) 199225. MR 2214145 (2007c:60050)
 4.
 R. Bañuelos, T. Kulczycki and Bartłomiej Siudeja, On the trace of symmetric stable processes on Lipschitz domains, J. Funct. Analysis 257(10) (2009) 33293352. MR 2568694
 5.
 F. A. Berezin, Covariant and contravariant symbols of operators, Izv. Akad. Nauk SSSR Ser. Mat. 36 (1972) 11341167. MR 0350504 (50:2996)
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 R. Blumenthal and R. Getoor, The asymptotic distribution of the eigenvalues for a class of Markov operators, Pacific J. Math. 9 (1959) 399408. MR 0107298 (21:6023)
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 E. M. Harrell II and L. Hermi, Differential inequalities for Riesz means and Weyltype bounds for eigenvalues, J. Funct. Analysis 254 (2008) 31733191. MR 2418623 (2009f:47067)
 8.
 E. M. Harrell II and S. Yildirim Yolcu, Eigenvalue inequalities for KleinGordon operators, J. Funct. Analysis, 256(12) (2009) 39773995. MR 2521917
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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.yildirimyolcu@gcsu.edu
DOI:
http://dx.doi.org/10.1090/S000299392010104197
Keywords:
Fractional Laplacian,
Weyl law,
universal bounds,
KleinGordon operator,
BerezinLiYau 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
