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 consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrodinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.
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.
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.
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.
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 consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Polyas conjecture is not true in the presence of a magnetic field.
