Green's functions for solving differential equations, in non-boundary value problems in near-field optics and in quantum transport through point contacts

TL;DR

This paper introduces Green's functions as a versatile tool for solving differential equations in physics, demonstrating their application in near-field optics and quantum transport, and providing numerical methods for practical calculations.

Contribution

It presents a unified approach to constructing Green's functions for inhomogeneous systems and applies this to electrodynamics and quantum transport, including numerical procedures.

Findings

Numerical method for near-field wave-equation in scattering setups

Green's function formalism for quantum transport in point contacts

Connection between Green's functions and physical observables like density of states

Abstract

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.