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Register a new userGeometrical Versions of improved Berezin-Li-Yau Inequalities
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We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in $R^d$, $d geq 2$. In particular, we derive upper bounds on Riesz means of order $sigma geq 3/2$, 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.
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In this paper we study the eigenvalue sums of Dirichlet Laplacians on bounded domains. Among our results we establish an improvement of the Li-Yau bound in the presence of a constant magnetic field.
We give an improvement of sharp Berezin type bounds on the Riesz means $sum_k(Lambda-lambda_k)_+^sigma$ of the eigenvalues $lambda_k$ of the Dirichlet Laplacian in a domain if $sigmageq 3/2$. It contains a correction term of the order of the standard second term in the Weyl asymptotics. The result is based on an application of sharp Lieb-Thirring inequalities with operator valued potential to spectral estimates of the Dirichlet Laplacian in domains.
We improve the Berezin-Li-Yau inequality in dimension two by adding a positive correction term to its right-hand side. It is also shown that the asymptotical behaviour of the correction term is almost optimal. This improves a previous result by Melas.
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