## A lower bound for sums of eigenvalues of the Laplacian

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- by Antonios D. Melas PDF
- Proc. Amer. Math. Soc.
**131**(2003), 631-636 Request permission

## Abstract:

Let $\lambda _{k}(\Omega )$ be the $k$th Dirichlet eigenvalue of a bounded domain $\Omega$ in $\mathbb {R}^{n}$. According to Weyl’s asymptotic formula we have \[ \lambda _{k}(\Omega )\thicksim C_{n}(k/V(\Omega ))^{2/n}.\] The optimal in view of this asymptotic relation lower estimate for the sums $\sum _{j=1}^{k}\lambda _{j}(\Omega )$ 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 $k$ that depends on the ratio of the volume to the moment of inertia of $\Omega$.

## References

- Pawel Kröger,
*Estimates for sums of eigenvalues of the Laplacian*, J. Funct. Anal.**126**(1994), no. 1, 217–227. MR**1305068**, DOI 10.1006/jfan.1994.1146 - 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,
*The number of bound states of one-body Schroedinger operators and the Weyl problem*, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 241–252. MR**573436** - G. Pólya,
*On the eigenvalues of vibrating membranes*, Proc. London Math. Soc. (3)**11**(1961), 419–433. MR**129219**, DOI 10.1112/plms/s3-11.1.419 - Barry Simon,
*Analysis with weak trace ideals and the number of bound states of Schrödinger operators*, Trans. Amer. Math. Soc.**224**(1976), no. 2, 367–380. MR**423128**, DOI 10.1090/S0002-9947-1976-0423128-X - Robert S. Strichartz,
*Estimates for sums of eigenvalues for domains in homogeneous spaces*, J. Funct. Anal.**137**(1996), no. 1, 152–190. MR**1383015**, DOI 10.1006/jfan.1996.0043

## Additional Information

**Antonios D. Melas**- Affiliation: Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece
- MR Author ID: 311078
- Email: amelas@math.uoa.gr
- Received by editor(s): August 28, 2001
- Published electronically: September 25, 2002
- Communicated by: Bennett Chow
- © Copyright 2002 American Mathematical Society
- 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
- MathSciNet review: 1933356