论文标题
具有可总结矩阵电势的非压缩量子图
Non-compact quantum graphs with summable matrix potentials
论文作者
论文摘要
令$ \ mathcal {g} $为有限的边缘的公制非划分的图形。本文的主要对象是$ l^2（\ Mathcal {g}; \ Mathbb {c}^m）$ hamiltonian $ {\ bf h}_α$与矩阵sturm-liouville表达式和边界delta-type条件相关联。假设潜在矩阵是可以总结的，并且应用边界三重态的技术和相应的Weyl函数，我们表明，哈密顿式$ {\ bf h}_α$的奇异连续频谱以及Sturm-liouville表达的任何其他自我相互接合实现。我们还指示图表上的条件，以确保$ {\ bf H}_α$的正聚会的纯绝对连续性。在潜在矩阵的额外条件下，获得了$ {\ bf h}_α$的负特征值数量的Bargmann型估计。此外，对于星形$ \ MATHCAL {G} $，找到了对$ \ {{\ bf h}_α，{\ bf h} _d \} $的散射矩阵的公式
Let $\mathcal{G}$ be a metric noncompact connected graph with finitely many edges. The main object of the paper is the Hamiltonian ${\bf H}_α$ associated in $L^2(\mathcal{G};\mathbb{C}^m)$ with a matrix Sturm-Liouville expression and boundary delta-type conditions at each vertex. Assuming that the potential matrix is summable and applying the technique of boundary triplets and the corresponding Weyl functions, we show that the singular continuous spectrum of the Hamiltonian ${\bf H}_α$ as well as any other self-adjoint realization of the Sturm-Liouville expression is empty. We also indicate conditions on the graph ensuring pure absolute continuity of the positive part of ${\bf H}_α$. Under an additional condition on the potential matrix, a Bargmann-type estimate for the number of negative eigenvalues of ${\bf H}_α$ is obtained. Additionally, for a star graph $\mathcal{G}$ a formula is found for the scattering matrix of the pair $\{{\bf H}_α, {\bf H}_D\}$, where ${\bf H}_D$ is the Dirichlet operator on $\mathcal{G}$.
