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

Abstract

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

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References in corpus (1)

• The Weyl formula for planar annuli