具有分形边界的区域的Kuznecov公式
Kuznecov formulas for domains with fractal boundary
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中文总结 AI 辅助
本文证明了具有分形边界的紧致黎曼流形上区域的特征函数积分平方和的Kuznecov公式，给出了余项为$\lambda^{s-d}$的精确渐近，并识别了由边界穿越体积控制的两项展开，区分了格点与非格点情形。
中文摘要 AI 辅助
设$(M,g)$为无边界的紧致$d$维黎曼流形，且设$\u007b e_j\u007d_{j=0}^\infty$为具有频率$\{\lambda_j\}_{j=0}^\infty$的拉普拉斯特征函数的正交基。对于区域$\Omega\subset M$，考虑$$ N_\Omega(\lambda) = \sum_{\lambda_j\leq\lambda} \left|\int_\Omega e_j\\,dV_g\right|^2. $$ 假设$\partial\Omega$是$s$-Ahlfors正则的，其中$s\in{[d-1,d)}$，且$\Omega$满足两侧的corkscrew条件。我们证明$$ N_\Omega(\lambda) = \operatorname{vol}(\Omega)+O\bigl(\lambda^{s-d}\bigr), $$ 并且该余项是精确的。我们进一步识别控制第二项的几何量：当边界具有自然的$s$维穿越体积$\mathcal V^s(\partial\Omega)$时，精确地存在$\lambda^{s-d}$阶的渐近展开，在这种情况下$$ N_\Omega(\lambda) = \operatorname{vol}(\Omega) - C_{d,s}\\, \mathcal V^s(\partial\Omega)\\, \lambda^{s-d} + o\bigl(\lambda^{s-d}\bigr). $$ 对于可容许的自相似边界，在非格点情形下穿越体积存在，因此上述两项渐近展开成立。在格点情形下，二阶行为则由对数周期轮廓所控制。
英文摘要
Let $(M,g)$ be a compact $d$-dimensional Riemannian manifold without boundary, and let $\{e_j\}_{j=0}^\infty$ be an orthonormal basis of Laplace eigenfunctions with frequencies $\{λ_j\}_{j=0}^\infty$. For a domain $Ω\subset M$, consider $$ N_Ω(λ) = \sum_{λ_j\leqλ} \left|\int_Ωe_j\,dV_g\right|^2. $$ Suppose that $\partialΩ$ is $s$-Ahlfors regular, with $s\in{[d-1,d)}$, and that $Ω$ satisfies a two-sided corkscrew condition. We prove that $$ N_Ω(λ) = \operatorname{vol}(Ω)+O\bigl(λ^{s-d}\bigr), $$ and that this remainder is sharp. We further identify the geometric quantity governing the second term: an exact asymptotic of order $λ^{s-d}$ holds precisely when the boundary admits a natural $s$-dimensional crossing volume $\mathcal V^s(\partialΩ)$, in which case $$ N_Ω(λ) = \operatorname{vol}(Ω) - C_{d,s}\, \mathcal V^s(\partialΩ)\, λ^{s-d} + o\bigl(λ^{s-d}\bigr). $$ For admissible self-similar boundaries, the crossing volume exists in the non-lattice case, and hence the above two-term asymptotic holds. In the lattice case, the second-order behavior is instead governed by a log-periodic profile.
发表机构
- School of Mathematical Sciences, Zhejiang University（浙江大学数学科学学院）
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