Improved Berezin-Li-Yau inequalities with a remainder term

Abstract

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 $\sigma \ge 3/2$. 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.

Dedicated to M. Sh. Birman on the occasion of his 80th birthday