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

Guo, Jingwei

doi:10.1090/proc/15343

A note on the Weyl formula for balls in $\mathbb {R}^d$
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Proc. Amer. Math. Soc. 149 (2021), 1663-1675 Request permission

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 \mathrm {vol}(\mathscr {B})\mu ^d-C’_d\mathrm {vol}(\partial \mathscr {B})\mu ^{d-1}+O\left (\mu ^{d-2+\frac {131}{208}}(\log \mu )^{\frac {18627}{8320}}\right ) \end{equation*} as $\mu \rightarrow \infty$.

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