Green's function approach for quantum graphs: an overview

Fabiano M. Andrade, Alexandre G. M. Schmidt, Eduardo Vicentini, Bin K. Cheng, Marcos G. E. da Luz

Published 2016-01-05Version 1

Here we review the many interesting 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. Such 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 coefficients) defined on each one of the graph vertices. So, 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. This 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, like scattering solutions for the former and eigenspectrum and eigenstates for the latter, also addressing quasi-bound states. Concrete examples, like cube, binary trees and Sierpi\'nski-like, topologies are considered. Along the work, possible distinct applications using the Green's function methods for quantum graphs are outlined.