Differential inequalities for Riesz means and Weyl-type bounds for eigenvalues

Abstract

We derive differential inequalities and difference inequalities for Riesz means of eigenvalues of the Dirichlet Laplacian, R_σ(z) := \sum_k{(z -λ_k)_+^σ}. Here ${{\lambda }_k}_{k=1}^{\infty }$ are the ordered eigenvalues of the Laplacian on a bounded domain $\Omega \subset {\mathbb{R}}^d$, and $x_{+}:=\max (0,x)$ denotes the positive part of the quantity $x$. As corollaries of these inequalities, we derive Weyl-type bounds on ${\lambda }_k$, on averages such as $\overline{{\lambda }_k}:=\frac{1}{k}{\sum }_{\ell \le k}{\lambda }_{\ell }$, and on the eigenvalue counting function. For example, we prove that for all domains and all $k\ge j\frac{1+\frac{d}{2}}{1+\frac{d}{4}}$, {\bar{λ_{k}}}/{\bar{λ_{j}}} \le 2 (\frac{1+\frac d 4}{1+\frac d 2})^{1+\frac 2 d}({\frac k j})^{\frac 2 d}.

21 pages, 3 figures

References in corpus (1)

• On Harrell-Stubbe Type Inequalities for the Discrete Spectrum of a Self-Adjoint Operator