In this paper, we discuss the concept of quantum graphs with transparent vertices by considering the case where the graph interacts with an external time-independent field. In particular, we address the problem of transparent boundary conditions for quantum graphs, building on previous work on transparent boundary conditions for the stationary Schrodinger equation on a line. Physically relevant constraints making the vertex transparent under boundary conditions in the form of (weight) continuity and Kirchhoff rules are derived using two methods, the scattering approach and transparent boundary conditions for the time-independent Schrodinger equation. The latter is derived by extending the transparent boundary condition concept to the time-independent Schrodinger equation on driven quantum graphs. We also discuss how the eigenvalues and eigenfunctions of a quantum graph are influenced not only by its topology, but also by the shape(type) of a potential when an external field is involved.

翻译：本文通过考虑图与外部时间无关场相互作用的情形，讨论了具有透明顶点的量子图概念。具体而言，我们基于先前关于直线上一维定态薛定谔方程透明边界条件的工作，探讨了量子图透明边界条件问题。采用散射方法和时无关薛定谔方程透明边界条件两种方法，推导了使顶点在（权重）连续性及基尔霍夫规则边界条件下具有透明性的物理相关约束。后者通过将透明边界条件概念扩展至驱动量子图的时无关薛定谔方程推导得出。我们还讨论了当引入外场时，量子图的本征值和本征函数不仅受拓扑结构影响，还受势能形状（类型）的影响。