论文标题
边界条件重要：在无限量子图的光谱上
Boundary Conditions Matter: On The Spectrum Of Infinite Quantum Graphs
论文作者
论文摘要
我们研究了在无限公制图，所谓的弗里德里奇和诺伊曼扩展上的两个明显的自我参与实现的频谱，几何形状和边界条件之间的相互作用。我们引入了一个新的标准，以解决分解的紧凑性，并将其应用于一类无限度量图中从纯离散到非空的基本频谱的过渡，这种现象似乎在无限域中的欧几里得域上没有已知的拉普拉奇人的对应物。然后，在离散频谱的情况下，我们证明了特征值上的上限和下限，因此将以前仅在紧凑型设置中已知的界限扩展到无限图。例如，即使在紧凑型图上，我们的某些界限，例如在Inradius方面都是新的。
We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called Friedrichs and Neumann extensions. We introduce a new criterion for compactness of the resolvent and apply this to identify a transition from purely discrete to non-empty essential spectrum among a class of infinite metric graphs, a phenomenon that seems to have no known counterpart for Laplacians on Euclidean domains of infinite volume. In the case of discrete spectrum we then prove upper and lower bounds on eigenvalues, thus extending a number of bounds previously only known in the compact setting to infinite graphs. Some of our bounds, for instance in terms of the inradius, are new even on compact graphs.
