Improved Berezin-Li-Yau inequality and Kr\"oger inequality and consequences

TL;DR

This paper presents quantitative improvements to classical spectral inequalities, resolving an open question and demonstrating the existence of infinitely many eigenvalues satisfying Pólya's conjecture in certain dimensions.

Contribution

It provides the first quantitative improvements to Berezin-Li-Yau and Kröger inequalities, and addresses an open problem in spectral theory.

Findings

Improved bounds for Berezin-Li-Yau inequality.

Resolution of Weidl's open question on Kröger inequality.

Existence of infinitely many eigenvalues satisfying Pólya's conjecture in specific dimensions.

Abstract

We provide quantitative improvements to the Berezin-Li-Yau inequality and the Kr\"oger inequality, in $\mathbb{R}^n$, $n\ge 2$. The improvement on Kr\"oger's inequality resolves an open question raised by Weidl from 2006. The improvements allow us to show that, for any open bounded domains, there are infinite many Dirichlet eigenvalues satisfying P\'olya's conjecture if $n\ge 3$, and infinite many Neumann eigenvalues satisfying P\'olya's conjecture if $n\ge 5$ and the Neumann spectrum is discrete.