Closed form expressions for the Green's function of a quantum graph -- a scattering approach

TL;DR

This paper derives a closed-form expression for the Green's function on finite quantum graphs using a scattering approach, simplifying previous methods and addressing bound state singularities.

Contribution

It introduces a three-step procedure to obtain explicit Green's functions for quantum graphs, generalizes prior work, and handles singularities from bound states.

Findings

Explicit Green's function expressed as a sum over eigenvalues

Regularization of singularities related to bound states

Equivalence of explicit expression to spectral decomposition

Abstract

In this work we present a three step procedure for generating a closed form expression of the Green's function on both closed and open finite quantum graphs with general self-adjoint matching conditions. We first generalize and simplify the approach by Barra and Gaspard [Barra F and Gaspard P 2001, Phys. Rev. E {\bf 65}, 016205] and then discuss the validity of the explicit expressions. For compact graphs, we show that the explicit expression is equivalent to the spectral decomposition as a sum over poles at the discrete energy eigenvalues with residues that contain projector kernel onto the corresponding eigenstate. The derivation of the Green's function is based on the scattering approach, in which stationary solutions are constructed by treating each vertex or subgraph as a scattering site described by a scattering matrix. The latter can then be given in a simple closed form from which the Green's function is derived. The relevant scattering matrices contain inverse operators which are not well defined for wave numbers at which bound states in the continuum exists. It is shown that the singularities in the scattering matrix related to these bound states or perfect scars can be regularised. Green's functions or scattering matrices can then be expressed as a sum of a regular and a singular part where the singular part contains the projection kernel onto the perfect scar.