A note on the Weyl formula for balls in ℝ^{𝕕},Proceedings of the American Mathematical Society

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A note on the Weyl formula for balls in ℝ^{𝕕}
Proceedings of the American Mathematical Society ( IF 0.8 ) Pub Date : 2021-02-12 , DOI: 10.1090/proc/15343
Jingwei Guo

Abstract:Let $\mathscr {B}=\{x\in \mathbb{R}^d : \vert x\vert<R \}$ ( $d\geq 3$ ) be a ball. We consider the Dirichlet Laplacian associated with $\mathscr {B}$ and prove that its eigenvalue counting function has an asymptotics

$\displaystyle \mathscr {N}_\mathscr {B}(\mu )=C_d \textnormal {vol}(\mathscr {B... ...1}+O\left (\mu ^{d-2+\frac {131}{208}}(\log \mu )^{\frac {18627}{8320}}\right )$

as $\mu \rightarrow \infty$ .

中文翻译：

关于ℝ^ {𝕕}中球的Weyl公式的注释

摘要：让（）成为一个球。我们考虑Dirichlet Laplacian与关联，并证明其特征值计数函数具有渐近性 $\ mathscr {B} = \ {x \ in \ mathbb {R} ^ d：\ vert x \ vert <R \}$ $d \ geq 3$ $\ mathscr {B}$

$\ displaystyle \ mathscr {N} _ \ mathscr {B}（\ mu）= C_d \ textnormal {vol}（\ mathscr {B ... ... 1} + O \ left（\ mu ^ {d-2 + \ frac {131} {208}}（\ log \ mu）^ {\ frac {18627} {8320}} \ right）$

作为。 $\ mu \ rightarrow \ infty$

更新日期：2021-04-12

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