发表机构
Faculty of Mathematics and Computer Science, University of Science, Vietnam National University; AISIA Lab（越南国立大学理科大学数学与计算机科学系; AISIA实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究证明等边三角形在狄利克雷、诺伊曼边界条件下的拉普拉斯本征函数一致非局域化，且诺伊曼估计在0≤σ≤σ₀的罗宾边界条件下仍一致成立，还区分了小罗宾定理与全局谱复杂度问题。
AI 中文摘要
我们研究了等边三角形上完全拉普拉斯本征空间在狄利克雷、诺伊曼与罗宾边界条件下的一致非局域化，特别关注诺伊曼问题在罗宾扰动下的稳定性。作为一个固定参考估计，我们给出了一个自包含的证明：即使存在任意大的算术重数，每个正测度的可观测集V⊂T都能捕获狄利克雷或诺伊曼本征空间中每个向量的L²质量中与频率无关的正比例。该证明将三角形展开为平坦环面，并结合二维傅里叶簇论证与Jarník型格点引理。我们的主要扰动结果表明，在罗宾边界条件下，诺伊曼估计在0≤σ≤σ₀范围内一致成立。McCartin的精确罗宾参数化产生了谱指标一致的模态空间估计，而Rudnick和Wigman的小参数简单性定理阻止了不同的去对称化谱类合并。最后，对于任意固定的σ>0，我们证明罗宾模态类的有界重合复杂度足以通过多元Turán–Nazarov不等式实现一致的本征空间级观测。因此，无条件的小罗宾定理与一般固定罗宾参数下剩余的全局谱复杂度问题被区分开来。
英文摘要
We study the high-energy spectral effect of Robin boundary perturbations on complete Laplace eigenspaces and its consequences for eigenfunction non-localization. Highly degenerate Neumann levels generally split under Robin perturbation, but this splitting need not produce a simple spectrum: distinct modal classes may still contribute to the same Robin eigenvalue. We prove that, for every fixed positive Robin parameter, the number of modal classes contributing to any one eigenvalue is uniformly bounded over the spectrum on the equilateral triangle and on every rectangle with rational squared aspect ratio. On the equilateral triangle, for all sufficiently small Robin parameters, the Neumann complete-eigenspace observation estimate persists with a constant uniform both in the eigenvalue and in the boundary parameter. For an arbitrary fixed positive Robin parameter, a high-frequency shell-splitting analysis gives a spectrum-wide bound on the number of modal classes contributing to any one Robin eigenvalue. In the nonsquare rectangular case, the corresponding splitting is governed by a strongly convex profile on weighted quadratic shells. Since each modal class has uniformly boundedplane-wave complexity, the spectral complexity bound implies observation on every measurable set of positive measure through a multidimensional Turán--Nazarov inequality, without a frequency-separation assumption. Thus, spectral simplicity is not required for complete-eigenspace non-localization: uniformly bounded spectral coincidence complexity is sufficient.
Comments60 pages. Submitted for publication
