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

Abstract

Let B = { x ∈ R d : | x | > R } \mathscr {B}=\{x\in \mathbb {R}^d : |x|>R \} ( d ≥ 3 d\geq 3 ) be a ball. We consider the Dirichlet Laplacian associated with B \mathscr {B} and prove that its eigenvalue counting function has an asymptotics N B ( μ ) = C d v o l ( B ) μ d − C d ′ v o l ( ∂ B ) μ d − 1 + O ( μ d − 2 + 131 208 ( log ⁡ μ ) 18627 8320 ) \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 μ → ∞ \mu \rightarrow \infty .