## An improvement to a Berezin-Li-Yau type inequality

HTML articles powered by AMS MathViewer

- by Selma Yıldırım Yolcu
- Proc. Amer. Math. Soc.
**138**(2010), 4059-4066 - DOI: https://doi.org/10.1090/S0002-9939-2010-10419-7
- Published electronically: May 18, 2010
- PDF | 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 ${\mathbb R}^d$.## References

- N. I. Akhiezer and I. M. Glazman,
*Theory of linear operators in Hilbert space. Vol. I*, Monographs and Studies in Mathematics, vol. 9, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1981. Translated from the third Russian edition by E. R. Dawson; Translation edited by W. N. Everitt. MR**615736** - Rodrigo Bañuelos and Tadeusz Kulczycki,
*The Cauchy process and the Steklov problem*, J. Funct. Anal.**211**(2004), no. 2, 355–423. MR**2056835**, DOI 10.1016/j.jfa.2004.02.005 - Rodrigo Bañuelos and Tadeusz Kulczycki,
*Eigenvalue gaps for the Cauchy process and a Poincaré inequality*, J. Funct. Anal.**234**(2006), no. 1, 199–225. MR**2214145**, DOI 10.1016/j.jfa.2005.11.016 - Rodrigo Bañuelos, Tadeusz Kulczycki, and Bartłomiej Siudeja,
*On the trace of symmetric stable processes on Lipschitz domains*, J. Funct. Anal.**257**(2009), no. 10, 3329–3352. MR**2568694**, DOI 10.1016/j.jfa.2009.06.037 - F. A. Berezin,
*Covariant and contravariant symbols of operators*, Izv. Akad. Nauk SSSR Ser. Mat.**36**(1972), 1134–1167 (Russian). MR**0350504** - R. M. Blumenthal and R. K. Getoor,
*The asymptotic distribution of the eigenvalues for a class of Markov operators*, Pacific J. Math.**9**(1959), 399–408. MR**107298** - Evans M. Harrell II and Lotfi Hermi,
*Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues*, J. Funct. Anal.**254**(2008), no. 12, 3173–3191. MR**2418623**, DOI 10.1016/j.jfa.2008.02.016 - Evans M. Harrell II and Selma Yıldırım Yolcu,
*Eigenvalue inequalities for Klein-Gordon operators*, J. Funct. Anal.**256**(2009), no. 12, 3977–3995. MR**2521917**, DOI 10.1016/j.jfa.2008.12.008 - Alexei A. Ilyin,
*Lower bounds for the spectrum of the Laplace and Stokes operators*, Discrete and Continuous Dynamical Systems, Volume 28, Number 1, September 2010, pp. 131–146. arXiv:0909.2818v1. - Hynek Kovařík, Semjon Vugalter, and Timo Weidl,
*Two-dimensional Berezin-Li-Yau inequalities with a correction term*, Comm. Math. Phys.**287**(2009), no. 3, 959–981. MR**2486669**, DOI 10.1007/s00220-008-0692-1 - Ari Laptev and Timo Weidl,
*Recent results on Lieb-Thirring inequalities*, Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000) Univ. Nantes, Nantes, 2000, pp. Exp. No. XX, 14. MR**1775696** - Peter Li and Shing Tung Yau,
*On the Schrödinger equation and the eigenvalue problem*, Comm. Math. Phys.**88**(1983), no. 3, 309–318. MR**701919** - Elliott H. Lieb and Michael Loss,
*Analysis*, 2nd ed., Graduate Studies in Mathematics, vol. 14, American Mathematical Society, Providence, RI, 2001. MR**1817225**, DOI 10.1090/gsm/014 - Antonios D. Melas,
*A lower bound for sums of eigenvalues of the Laplacian*, Proc. Amer. Math. Soc.**131**(2003), no. 2, 631–636. MR**1933356**, DOI 10.1090/S0002-9939-02-06834-X - Timo Weidl,
*Improved Berezin-Li-Yau inequalities with a remainder term*, Spectral theory of differential operators, Amer. Math. Soc. Transl. Ser. 2, vol. 225, Amer. Math. Soc., Providence, RI, 2008, pp. 253–263. MR**2509788**, DOI 10.1090/trans2/225/17

## Bibliographic 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
- 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

Dedicated: This paper is dedicated to Professor Evans M. Harrell