Summary: "We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in ℝd, d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.
Ünder certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.''
%0 Journal Article
%1 geisinger2011geometrical
%A Geisinger, Leander
%A Laptev, Ari
%A Weidl, Timo
%D 2011
%J Journal of Spectral Theory 1.
%K Berezin-Li-Yau Geisinger geometrical iadm weidl
%N 1
%P 87-109
%T Geometrical versions of improved Berezin-Li-Yau inequalities.
%U https://arxiv.org/pdf/1010.2683.pdf
%X Summary: "We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in ℝd, d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.
Ünder certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.''
@article{geisinger2011geometrical,
abstract = {Summary: "We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in ℝd, d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.
"Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li-Yau inequality.''},
added-at = {2021-12-27T08:05:01.000+0100},
author = {Geisinger, Leander and Laptev, Ari and Weidl, Timo},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/23008ae37c86d925e359035d86dddfb53/elkepeter},
interhash = {12f1a0a5525d80cffc97f02fc44337c9},
intrahash = {3008ae37c86d925e359035d86dddfb53},
journal = {Journal of Spectral Theory 1.},
keywords = {Berezin-Li-Yau Geisinger geometrical iadm weidl},
language = {Englisch},
number = 1,
pages = {87-109},
timestamp = {2023-04-21T13:36:12.000+0200},
title = {Geometrical versions of improved Berezin-Li-Yau inequalities.},
url = {https://arxiv.org/pdf/1010.2683.pdf},
year = 2011
}
