A note on the Weyl formula for balls in $\mathbb{R}^d$

TL;DR

This paper refines the asymptotic formula for the eigenvalue counting function of the Dirichlet Laplacian on a ball in \\mathbb{R}^d, providing a sharper error term for large eigenvalues.

Contribution

It establishes a precise asymptotic expansion with an improved error estimate for the eigenvalue counting function of the Dirichlet Laplacian on a ball.

Findings

Asymptotic formula for eigenvalue counting function with explicit constants

Refined error term involving \\mu^{d-2+131/208} and a logarithmic factor

Enhanced understanding of spectral asymptotics for spherical domains

Abstract

Let $\mathscr{B}=\{x\in\mathbb{R}^d : |x|<R \}$ ($d\geq 3$) be a ball. We consider the Dirichlet Laplacian associated with $\mathscr{B}$ and prove that its eigenvalue counting function has an asymptotics \begin{equation*} \mathscr{N}_\mathscr{B}(\mu)=C_d vol(\mathscr{B})\mu^d-C'_d vol(\partial \mathscr{B})\mu^{d-1}+O\left(\mu^{d-2+\frac{131}{208}}(\log \mu)^{\frac{18627}{8320}}\right) \end{equation*} as $\mu\rightarrow \infty$.