Green's function approach for quantum graphs: an overview

Here we review the many aspects and distinct phenomena associated to quantum dynamics on general graph structures. For so, we discuss such class of systems under the energy domain Green's function ($G$) framework. This approach is particularly interesting because $G$ can be written as a sum over classical-like paths, where local quantum effects are taking into account through the scattering matrix amplitudes (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the {\em exact} $G$ has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in details here) always can be cast into a closed analytic expression. It allows to solve exactly arbitrary large (although finite) graphs in a recursive and fast way. Using the Green's function method, we survey many properties for open and closed quantum graphs as scattering solutions for the former and eigenspectrum and eigenstates for the latter, also considering quasi-bound states. Concrete examples, like cube, binary trees and Sierpi\'{n}ski-like topologies are presented. Along the work, possible distinct applications using the Green's function methods for quantum graphs are outlined.Comment: 54 pages, 24 figures, 1 table. Two sections expanded + minor modifications. One appendix added. References added. To appear in Physics Report