We consider the Robin Laplacian in the domains Ω and Ωε, ε > 0, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient a is large enough, the spectrum of the problem in Ω is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain Ωε is discrete. However, our results reveal the strange behaviour of the discrete spectrum as the blunting parameter ε tends to 0: we construct asymptotic forms of the eigenvalues and detect families of ‘hardly movable’ and ‘plummeting’ ones. The first type of the eigenvalues do not leave a small neighbourhood of a point for any small ε > 0 while the second ones move at a high rate O(| ln ε|) downwards along the real axis ℝ to −∞. At the same time, any point λ ∈ ℝ is a ‘blinking eigenvalue’, i.e., it belongs to the spectrum of the problem in Ωε almost periodically in the | ln ε|-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

中文翻译：

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尖角域中罗宾拉普拉斯算子的直线下降和闪烁特征值

我们考虑域 Ω 和 Ω 中的 Robin Laplacianε,ε> 0，分别有尖尖和钝尖。假设罗宾系数一种足够大，已知问题在 Ω 中的谱是残差的，并且覆盖了整个复平面，但相反，Lipschitz 域 Ω 中的谱ε是离散的。然而，我们的结果揭示了离散谱作为钝化参数的奇怪行为ε趋于 0：我们构造特征值的渐近形式并检测“几乎不可移动”和“直线下降”的族。第一种特征值不会为任何小的点留下一个小的邻域ε> 0，而第二个以高速率移动○(| lnε|) 沿实轴 ℝ 向下到 -∞。同时，任意一点 λ ∈ ℝ 是一个“闪烁特征值”，即属于问题的谱 Ωε几乎定期在| lnε|-规模。除了标准的光谱理论，我们使用降维和自伴随扩展的技术来获得这些结果。