Universal Bounds for Traces of the Dirichlet Laplace Operator

Details Universal Bounds for Traces of the Dirichlet Laplace Operator Universal Bounds for Traces of the Dirichlet Laplace Operator

2009-09-07

arxiv.org/abs/0909.1266v2

J. Lond. Math. Soc. 82 (2010), no. 2, 395-419

Physics

Mathematical Physics

Scientific paper

10.1112/jlms/jdq033

We derive upper bounds for the trace of the heat kernel $Z(t)$ of the Dirichlet Laplace operator in an open set $\Omega \subset \R^d$, $d \geq 2$. In domains of finite volume the result improves an inequality of Kac. Using the same methods we give bounds on $Z(t)$ in domains of infinite volume. For domains of finite volume the bound on $Z(t)$ decays exponentially as $t$ tends to infinity and it contains the sharp first term and a correction term reflecting the properties of the short time asymptotics of $Z(t)$. To prove the result we employ refined Berezin-Li-Yau inequalities for eigenvalue means.

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