平面狄利克雷谱中均匀分离的普适失效
Generic failure of uniform separation in planar Dirichlet spectra
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中文总结 AI 辅助
该研究证明平面狄利克雷谱中均匀分离是普适例外，一般区域具简单谱且连续间隙下极限为0，用Šverák定理和局部手术完成证明。
中文摘要 AI 辅助
有界平面区域能否具有带有均匀分离连续特征值的简单狄利克雷谱？二维是临界维度：Weyl定律既允许均匀分离，也允许任意小的间隙。我们证明，在自然的粗糙区域语境中，即使去除重数，均匀分离仍是例外。设$D/\text{subset}//\text{R}^2$为有界区域，对$/\text{ell}/\text{geq}/1$，令$/\text{C}_/\text{ell}(D)$为满足$/\text{overline}/D/\text{setminus}/\text{Omega}$至多有$/\text{ell}$个连通分支的非空连通开集$/\text{Omega}/\text{subset}/D$构成的空间，赋予互补-豪斯多夫拓扑。我们证明$/\text{C}_/\text{ell}(D)$是完全可度量化的贝尔空间，且光滑区域在其中稠密。若$0</\text{nu}_1(/\text{Omega})</\text{nu}_2(/\text{Omega})</\text{cdots}$为不同的狄利克雷特征值，我们的主要结果表明，集合$/\text{\{}/\text{Omega}/\text{in}/\text{C}_/\text{ell}(D):/\text{inf}/\text{\textunderscore}/\text{m}/\text{geq}/1/\text{bigl}(/\text{nu}/\text{\textunderscore}/\text{m+1}(/\text{Omega})-/\text{nu}/\text{\textunderscore}/\text{m}(/\text{Omega})/\text{bigr})=0/\text{\text{\textbackslash}\text{rbrace}}$是剩余集，该结论无需简单性假设。将其与我们将Micheletti的经典普适简单性定理转移到$/\text{C}_/\text{ell}(D)$的结果相结合，表明一般区域具有简单谱且连续间隙的下极限为0。证明使用Šverák的平面谱连续性定理和植入任意高对接近的连续不同特征值的局部手术。
英文摘要
Can a bounded planar domain have a simple Dirichlet spectrum with uniformly separated consecutive eigenvalues? Dimension two is critical: Weyl's law permits both uniform separation and arbitrarily small gaps. We prove that uniform separation is nevertheless exceptional in a natural rough-domain setting, even after multiplicities are removed. Let $D\subset\mathbb{R}^2$ be a bounded domain and, for $\ell\geq 1$, let $\mathcal{C}_\ell(D)$ be the space of nonempty connected open sets $Ω\subset D$ such that $\overline{D}\setminusΩ$ has at most $\ell$ connected components, endowed with the complementary-Hausdorff topology. We prove that $\mathcal{C}_\ell(D)$ is completely metrizable and Baire, and that smooth domains are dense in it. If $0<ν_1(Ω)<ν_2(Ω)<\cdots$ are the distinct Dirichlet eigenvalues, our main result states that \[ \left\{Ω\in\mathcal{C}_\ell(D):\inf_{m\geq 1}\bigl(ν_{m+1}(Ω)-ν_m(Ω)\bigr)=0\right\} \] is residual. This statement requires no simplicity assumption. Combining it with our transfer of Micheletti's classical generic-simplicity theorem to $\mathcal{C}_\ell(D)$ shows that a generic domain has simple spectrum and consecutive gaps with zero lower limit. The proof uses Šverák's planar spectral continuity theorem and a local surgery that implants an arbitrarily high pair of close consecutive distinct eigenvalues.
