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Green’s function approach for quantum graphs: an overview. (English) Zbl 1359.81117

Summary: 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 taken into account through the scattering matrix elements (basically, transmission and reflection amplitudes) defined on each one of the graph vertices. Hence, the exact \(G\) has the functional form of a generalized semiclassical formula, which through different calculation techniques (addressed in detail 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 of 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 Sierpiński-like topologies are presented. Along the work, possible distinct applications using the Green’s function methods for quantum graphs are outlined.

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U05 \(2\)-body potential quantum scattering theory
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