arXiv 2301.059302023-01-14math.SPmath.AP

薄立方晶格中Dirichlet Laplacian的谱

Spectrum of the Dirichlet Laplacian in a thin cubic lattice

Lucas Chesnel, Sergei A. Nazarov

摘要：

我们给出了一个描述的Dirichlet拉普拉斯算子的频谱的下部在一个无界的三维周期性晶格制成的细棒（宽度$\varepsilon\ll1$），其中有一个正方形截面。该谱与由于狄利克雷边界条件而导致$\varepsilon$趋于零的全部去往$+\infty$的段的并集一致。我们发现，第一个频谱段是非常紧的，长度为$O(e^{-δ/\varepsilon})$，$δ>0$，而下一个频谱段的长度为$O(\varepsilon)$。要建立这些结果，我们需要详细研究的性质的狄利克雷拉普拉斯$A^Ω$的几何$Ω$通过放大的结区域的初始周期晶格。这个问题有它自己的兴趣和发挥对称性与最大-最小参数以及一个精心选择的弗里德里希不等式，我们证明了$A^Ω$有一个独特的特征值在其离散谱，产生的第一个谱段。此外，我们还证明了$A^Ω$不存在阈值共振，即在阈值频率下$A^Ω$不存在非平凡有界解.这意味着下一个谱段的晶格的正确1D模型是顶点处具有狄利克雷条件的图。我们还提出了数字来补充分析。

英文摘要：

We give a description of the lower part of the spectrum of the Dirichlet Laplacian in an unbounded 3D periodic lattice made of thin bars (of width $\varepsilon\ll1$) which have a square cross section. This spectrum coincides with the union of segments which all go to $+\infty$ as $\varepsilon$ tends to zero due to the Dirichlet boundary condition. We show that the first spectral segment is extremely tight, of length $O(e^{-δ/\varepsilon})$, $δ>0$, while the length of the next spectral segments is $O(\varepsilon)$. To establish these results, we need to study in detail the properties of the Dirichlet Laplacian $A^Ω$ in the geometry $Ω$ obtained by zooming at the junction regions of the initial periodic lattice. This problem has its own interest and playing with symmetries together with max-min arguments as well as a well-chosen Friedrichs inequality, we prove that $A^Ω$ has a unique eigenvalue in its discrete spectrum, which generates the first spectral segment. Additionally we show that there is no threshold resonance for $A^Ω$, that is no non trivial bounded solution at the threshold frequency for $A^Ω$. This implies that the correct 1D model of the lattice for the next spectral segments is a graph with Dirichlet conditions at the vertices. We also present numerics to complement the analysis.

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英文摘要：

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