量子图的格林函数与薛定谔方法的么正等价性
Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs
摘要：
在先前的工作[Andrade \textit{et al.}，Phys. Rep. \textbf{647}，1（2016）]中，证明了任意大（尽管有限）量子图的精确格林函数（GF）是作为散射路径的总和给出的，其中通过反射和透射散射振幅考虑了局部量子效应。为了处理一般的图，开发了两个简化过程：将路径重组为路径族和将大型图分离为子图。然而，对于具有复杂拓扑结构的不太对称的图，例如随机图，选择子图和路径族可能变得很麻烦。在这项工作中，一个更一般的程序来构建能量域GF的量子图的基础上，其邻接矩阵。这种新的结构使我们能够获得长期的行列式，解开散射薛定谔方法和格林函数方法之间的酉等价。它还使我们能够编写基于格林函数方法的跟踪公式。本构造的优点是它可以直接应用于任何图，从规则到随机拓扑。
英文摘要：
In a previous work [Andrade \textit{et al.}, Phys. Rep. \textbf{647}, 1 (2016)], it was shown that the exact Green's function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effects are taken into account through the reflection and transmission scattering amplitudes. To deal with general graphs, two simplifying procedures were developed: regrouping of paths into families of paths and the separation of a large graph into subgraphs. However, for less symmetrical graphs with complicated topologies as, for instance, random graphs, it can become cumbersome to choose the subgraphs and the families of paths. In this work, an even more general procedure to construct the energy domain GF for a quantum graph based on its adjacency matrix is presented. This new construction allows us to obtain the secular determinant, unraveling a unitary equivalence between the scattering Schrödinger approach and the Green's function approach. It also enables us to write a trace formula based on the Green's function approach. The present construction has the advantage that it can be applied directly for any graph, going from regular to random topologies.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
