论文标题
在带有大复合$δ$耦合的量子图拉普拉斯的特征值上
On the eigenvalues of quantum graph Laplacians with large complex $δ$ couplings
论文作者
论文摘要
我们研究Laplacian在紧凑型公制图上的位置，具有复杂的罗宾型顶点条件（也称为$Δ$条件）在某些或全部图形顶点上。我们将特征值渐近学分类为复杂的robin参数（s）偏见到$ \ mathbb {c} $中的$ \ infty $：对于每个顶点$ v $，用robin参数$α\ in \ mathbb in \ mathbb {c} $ for whe $ \ mathrm in \ mathrm in f in \ mathbb {一个发散的特征值，其表现为$-α^2/\ mathrm {deg} \，v^2 $，而所有其他特征值则保持在laplacian的频谱附近，其条件为$ v $；如果$ \ mathrm {re} \，α$仍然从下面进行边界，则所有特征值保持在Dirichlet Spectrum附近。我们的证明是基于对相应的Dirichlet到Neumann矩阵（Titchmarsh-Weyl M命令）的分析。我们还使用尖锐的痕量不平等现象来证明对数值范围的估计以及对操作员的光谱的估计，这使我们能够根据robin参数的真实部分和虚构的部分（S）来控制特征值的真实和虚构部分。
We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as $δ$ 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 $α\in \mathbb{C}$ for which $\mathrm{\Re}\,α\to -\infty$ sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like $-α^2/\mathrm{deg}\,v^2$, while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at $v$; if $\mathrm{Re}\,α$ 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).
