Driven transparent quantum graphs

Abstract 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 works on transparent boundary conditions for the stationary Schrödinger equation on a line. Physically relevant constraints that make the vertex transparent under boundary conditions in the form of (weight) continuity and Kirchhoff rules are derived using two approaches, where transparent boundary conditions for the time-independent Schrödinger equation are found. The latter is derived by extending the transparent boundary condition concept to the time-independent Schrödinger equation on driven quantum graphs. For the constant potential, these constraints are also obtained using the scattering approach. 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.