Green's functions for solving differential equations, in non-boundary   value problems in near-field optics and in quantum transport through point   contacts
  This introduction to Green's functions is based on their role as kernels of differential equations. The procedures to construct solutions to a differential equation with an external source or with an inhomogeneity term are put together to derive the Dyson equation for the Green's function of the inhomogeneous system. Very different areas of physics such as, for example, electrodynamics and quantum transport, can profit from this Green's function formalism. The fundamental homogeneous-medium Green's tensor of electrodynamics is deduced from the field of a dipole. Based upon that a numerical procedure is presented to solve the wave-equation for the near-field in a scattering setup for arbitrary material distributions. The full inhomogeneous system's Green's function is not explicitly needed to get the fields, although it can be obtained by a very similar calculation and in optics can be interpreted as a density of states. It is demonstrated how the transport problem for two open free-electron gas reservoirs with arbitrary coupling can be solved by finding the system's Green's function. In this sense the article is an introduction on Green's functions for treating interaction. A very detailed discussion of the current formula is given on an elementary basis. 
Ursula Schr\"oter