Dirichlet边界条件下的分支量子波导：解耦情形
Branched quantum wave guides with Dirichlet boundary conditions: the decoupling case
摘要：
我们考虑一族开集$M_ε$，它相对于一个适当的参数$ε$收缩到一个图。在顶点邻域很小的附加假设下，证明了$M_ε$的适当移位的Dirichlet谱收敛于顶点处具有Dirichlet边界条件的图上的（微分）Laplacian的谱，即，一种不同边之间没有耦合的图算子。的小表示的混合特征值问题的顶点邻域上的第一个特征值的下界。下界由边缘邻域的第一横模给出。我们还允许弯曲的边缘，并表明，所有有界特征值收敛到一个拉普拉斯算子的频谱上的边缘与一个额外的潜在的来自曲率。
英文摘要：
We consider a family of open sets $M_ε$ which shrinks with respect to an appropriate parameter $ε$ to a graph. Under the additional assumption that the vertex neighbourhoods are small we show that the appropriately shifted Dirichlet spectrum of $M_ε$ converges to the spectrum of the (differential) Laplacian on the graph with Dirichlet boundary conditions at the vertices, i.e., a graph operator without coupling between different edges. The smallness is expressed by a lower bound on the first eigenvalue of a mixed eigenvalue problem on the vertex neighbourhood. The lower bound is given by the first transversal mode of the edge neighbourhood. We also allow curved edges and show that all bounded eigenvalues converge to the spectrum of a Laplacian acting on the edge with an additional potential coming from the curvature.
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