AI 中文总结
该研究针对平面可积多边形上的Dirichlet拉普拉斯本征函数，证明了矩形等几类多边形的一致$L^2$非局域化，得到定量阈值并建立稳定性，获得拟模式与窄谱簇的非局域化结果。
AI 中文摘要
我们研究平面可积多边形上Dirichlet拉普拉斯本征函数的一致$L^2$非局域化，特别关注谱简并性及其对观测集的定量依赖。对于正测度可测集$V/\u03a9$，定义$C_2(V;\u03a9):= \u2211_{\u03bb\u2208\u03c3(-\u0394_\u03a9)} \u2211_{0\u2260u\u2208E_\u03bb(\u03a9)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(\u03a9)}}$。我们证明矩形、等腰直角三角形、等边三角形和半等边三角形满足$C_2(V;\u03a9)>0$，且在整个本征空间上一致，与谱重数无关。对于矩形，我们得到定量改进：若$V$的反射延拓具有有限周长且$\u03b1=|V|/|\u03a9|$，则存在显式充分阈值$\u03bb_*(\u03a9,V)$，使得所有$\u03bb\u2265\u03bb_*(\u03a9,V)$的本征函数满足$\u03b1/2 \times (1-\frac{\u03c0\u03b1}{\u03c0\u03b1})^{1/2}$。我们进一步建立矩形分支在有界实值势下的稳定性，以及在受控谱缺陷下的稳定性，从而为足够精确的拟模式和窄谱簇得到相应的非局域化结果。
英文摘要
We study uniform $L^2$ non-localization of Dirichlet Laplacian eigenfunctions on planar integrable polygons, with particular emphasis on spectral degeneracy and quantitative dependence on the observation set. For a measurable set $V\subsetΩ$ of positive measure, define \[ C_2(V;Ω) := \inf_{λ\inσ(-Δ_Ω)} \inf_{0\neq u\in E_λ(Ω)} \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(Ω)}}. \] We prove that $C_2(V;Ω)>0$ for rectangles, isosceles right triangles, equilateral triangles, and hemi-equilateral triangles, uniformly over the complete eigenspaces and hence independently of spectral multiplicity. For rectangles, we obtain a quantitative refinement. If the reflected extension of $V$ has finite perimeter and $α=|V|/|Ω|$, we derive an explicit sufficient threshold $λ_*(Ω,V)$ such that every eigenfunction with $λ\geqλ_*(Ω,V)$ satisfies \[ \frac{\|u\|_{L^2(V)}}{\|u\|_{L^2(Ω)}} \geq \left[ \fracα{2} \left( 1-\frac{\sin(πα)}{πα} \right) \right]^{1/2}. \] We further establish stability under bounded real-valued potentials on the rectangular branch and under controlled spectral defects, yielding corresponding non-localization results for sufficiently accurate quasimodes and narrow spectral clusters.
Comments63 pages, 1 figure
