Improved Berezin-Li-Yau inequalities with a remainder term

TL;DR

This paper improves Berezin-Li-Yau inequalities for the eigenvalues of the Dirichlet Laplacian by adding a correction term, using sharp Lieb-Thirring inequalities, enhancing spectral bounds for domains.

Contribution

It introduces a refined inequality with a remainder term for Riesz means of Dirichlet Laplacian eigenvalues, based on advanced spectral analysis techniques.

Findings

Enhanced bounds on eigenvalue sums with correction terms

Application of Lieb-Thirring inequalities to domain spectral estimates

Refinement of Weyl asymptotics for eigenvalues

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 $\sigma\geq 3/2$. 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.