Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain

Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term in...

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Main Authors: Frank, Rupert L., Geisinger, Leander
Format: Text
Language:unknown
Published: arXiv 2011
Subjects:
Spectral Theory math.SP
Mathematical Physics math-ph
FOS Mathematics
FOS Physical sciences
Ari
Laplace
laptev
Online Access:https://dx.doi.org/10.48550/arxiv.1105.5182
https://arxiv.org/abs/1105.5182
id ftdatacite:10.48550/arxiv.1105.5182
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spelling ftdatacite:10.48550/arxiv.1105.5182 2023-05-15T17:07:16+02:00 Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain Frank, Rupert L. Geisinger, Leander 2011 https://dx.doi.org/10.48550/arxiv.1105.5182 https://arxiv.org/abs/1105.5182 unknown arXiv https://dx.doi.org/10.1142/9789814350365_0012 arXiv.org perpetual, non-exclusive license http://arxiv.org/licenses/nonexclusive-distrib/1.0/ Spectral Theory math.SP Mathematical Physics math-ph FOS Mathematics FOS Physical sciences article-journal Article ScholarlyArticle Text 2011 ftdatacite https://doi.org/10.48550/arxiv.1105.5182 https://doi.org/10.1142/9789814350365_0012 2022-04-01T14:28:34Z Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary. : 10 pages; dedicated to Ari Laptev on the occasion of his 60th birthday Text laptev DataCite Metadata Store (German National Library of Science and Technology) Ari ENVELOPE(147.813,147.813,59.810,59.810) Laplace ENVELOPE(141.467,141.467,-66.782,-66.782)
institution Open Polar
collection DataCite Metadata Store (German National Library of Science and Technology)
op_collection_id ftdatacite
language unknown
topic Spectral Theory math.SP
Mathematical Physics math-ph
FOS Mathematics
FOS Physical sciences
spellingShingle Spectral Theory math.SP
Mathematical Physics math-ph
FOS Mathematics
FOS Physical sciences
Frank, Rupert L.
Geisinger, Leander
Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
topic_facet Spectral Theory math.SP
Mathematical Physics math-ph
FOS Mathematics
FOS Physical sciences
description Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary. : 10 pages; dedicated to Ari Laptev on the occasion of his 60th birthday
format Text
author Frank, Rupert L.
Geisinger, Leander
author_facet Frank, Rupert L.
Geisinger, Leander
author_sort Frank, Rupert L.
title Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
title_short Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
title_full Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
title_fullStr Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
title_full_unstemmed Two-term spectral asymptotics for the Dirichlet Laplacian on a bounded domain
title_sort two-term spectral asymptotics for the dirichlet laplacian on a bounded domain
publisher arXiv
publishDate 2011
url https://dx.doi.org/10.48550/arxiv.1105.5182
https://arxiv.org/abs/1105.5182
long_lat ENVELOPE(147.813,147.813,59.810,59.810)
ENVELOPE(141.467,141.467,-66.782,-66.782)
geographic Ari
Laplace
geographic_facet Ari
Laplace
genre laptev
genre_facet laptev
op_relation https://dx.doi.org/10.1142/9789814350365_0012
op_rights arXiv.org perpetual, non-exclusive license
http://arxiv.org/licenses/nonexclusive-distrib/1.0/
op_doi https://doi.org/10.48550/arxiv.1105.5182
https://doi.org/10.1142/9789814350365_0012
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