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Asymptotics for the spectral heat function and bounds for integrals of Dirichlet eigenfunctions

Published online by Cambridge University Press:  14 November 2011

M. van den Berg  and
S. P. Watson
M. van den Berg
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
S. P. Watson
Affiliation:
School of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK
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Extract

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 4 , 1999 , pp. 841 - 854
Copyright
Copyright © Royal Society of Edinburgh 1999

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