We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $\delta$ conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to $\infty$ in $\mathbb{C}$: for each vertex $v$ with a Robin parameter $\alpha \in \mathbb{C}$ for which $\mathrm{\Re}\,\alpha \to -\infty$ sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like $-\alpha^2/\mathrm{deg}\,v^2$, while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at $v$; if $\mathrm{Re}\,\alpha$ remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh--Weyl M-functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s).

中文翻译：

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关于具有大复杂 $\delta$ 耦合的量子图拉普拉斯算子的特征值

我们研究了拉普拉斯算子在具有复杂罗宾型顶点条件（也称为 $\delta$ 条件）的紧凑度量图上的谱在部分或所有图顶点上的位置。我们将特征值渐近分类为复杂的 Robin 参数在 $\mathbb{C}$ 中发散到 $\infty$：对于每个顶点 $v$ 和一个 Robin 参数 $\alpha \in \mathbb{C}$其中 $\mathrm{\Re}\,\alpha \to -\infty$ 足够快，恰好存在一个发散特征值，其行为类似于 $-\alpha^2/\mathrm{deg}\,v^2$，而所有其他特征值都接近拉普拉斯算子的频谱，狄利克雷条件为 $v$；如果 $\mathrm{Re}\,\alpha$ 从下方保持有界，则所有特征值都保持在狄利克雷谱附近。我们的证明基于对相应 Dirichlet-to-Neumann 矩阵（Titchmarsh--Weyl M 函数）的分析。我们还使用锐痕型不等式来证明对数值范围的估计，从而证明对算子频谱的估计，这使我们能够根据 Robin 参数的实部和虚部来控制特征值的实部和虚部(s)。