Two new Weyl-type bounds for the Dirichlet Laplacian

TL;DR

This paper introduces two novel Weyl-type upper bounds for the eigenvalues of the Dirichlet Laplacian, leading to new lower bounds for its counting function based on geometric and spectral properties.

Contribution

The paper presents two new Weyl-type upper estimates for Dirichlet Laplacian eigenvalues, improving understanding of their spectral distribution and deriving explicit lower bounds for the counting function.

Findings

Derived two new Weyl-type upper bounds for eigenvalues.

Established lower bounds for the eigenvalue counting function.

Connected bounds to Bessel functions and geometric parameters.

Abstract

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following {\em lower} bounds for its counting function. For $\la\ge \la_1$, one has N(\la) > \dfrac{2}{n+2} \dfrac{1}{H_n} (\la-\la_1)^{n/2} \la_1^{-n/2}, and N(\la) > (\dfrac{n+2}{n+4})^{n/2} \dfrac{1}{H_n} (\la-(1+4/n) \la_1)^{n/2} \la_1^{-n/2}, where H_n=\dfrac{2 n}{j_{n/2-1,1}^2 J_{n/2}^2(j_{n/2-1,1})} is a constant which depends on $n$, the dimension of the underlying space, and Bessel functions and their zeros.