Green's function approach for quantum graphs: an overview

TL;DR

This paper reviews the Green's function approach to quantum graphs, highlighting its ability to provide exact solutions for complex structures by summing over classical-like paths with quantum effects incorporated.

Contribution

It presents a comprehensive overview of using Green's functions to analyze quantum graphs, including derivation techniques and applications to various topologies.

Findings

Exact solutions for large finite graphs are achievable.

Green's function can be expressed as a sum over classical paths with quantum corrections.

Applications include scattering, eigenstates, and quasi-bound states analysis.

Abstract

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.