A lower bound for sums of eigenvalues of the Laplacian

Author:
Antonios D. Melas

Journal:
Proc. Amer. Math. Soc. **131** (2003), 631-636

MSC (2000):
Primary 58G25; Secondary 35P15, 58G05

DOI:
https://doi.org/10.1090/S0002-9939-02-06834-X

Published electronically:
September 25, 2002

MathSciNet review:
1933356

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be the th Dirichlet eigenvalue of a bounded domain in . According to Weyl's asymptotic formula we have

The optimal in view of this asymptotic relation lower estimate for the sums has been proven by P.Li and S.T.Yau (

*Comm. Math. Phys.*

**88**(1983), 309-318). Here we will improve this estimate by adding to its right-hand side a term of the order of that depends on the ratio of the volume to the moment of inertia of .

**1.**P.Kröger: Estimates for sums of Eigenvalues of the Laplacian,*Jour. Funct. Anal.***126**(1994), 217-227. MR**95j:58173****2.**P.Li, S.T.Yau: On the Schrödinger equation and the eigenvalue problem,*Comm. Math. Phys.***88**(1983), 309-318. MR**84k:58225****3.**E.Lieb: The number of bound states of one-body Schrö ndinger operators and the Weyl problem,*Proc. Sym. Pure Math.***36**(1980), 241-252. MR**82i:35134****4.**G.Pólya: On the eigenvalues of vibrating membranes,*Proc. London Math. Soc.*(3)**11**(1961), 419-433. MR**23:B2256****5.**B.Simon: Weak trace ideals and the number of bound states of Schrödinger operators,*Trans. Amer. Math. Soc.***224**(1976), 367-380. MR**54:11109****6.**R.S.Strichartz: Estimates for sums of eigenvalues for domains in homogeneous spaces,*Jour. Funct. Anal.***137**(1996), 152-190. MR**97g:58172**

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Additional Information

**Antonios D. Melas**

Affiliation:
Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece

Email:
amelas@math.uoa.gr

DOI:
https://doi.org/10.1090/S0002-9939-02-06834-X

Received by editor(s):
August 28, 2001

Published electronically:
September 25, 2002

Communicated by:
Bennett Chow

Article copyright:
© Copyright 2002
American Mathematical Society