Green Functions  

For Differential Equations > s.a. fokker-planck equation; Propagator; wave equations.
$ Def: For a second-order linear differential operator L, the symmetric 2-point function G satisfying

L G(x, x') = δ(xx') ;

Notice that a given operator L has many Green functions, depending on the boundary conditions imposed on the solution.
* Applications: It is used to find solutions of the differential equation = j, given the source j and the boundary conditions on the field φ, i.e., to propagate the field; It is thus also called propagator.
* Specific types of equations: For the Laplacian L = ∇2, the Green function is \(G(x,x')\) = 1 / |xx'|; This applies to electrostatics and Newtonian gravity.
@ Specific types of equations: Haba JPA(04)ht, JMP(05)mp [strongly inhomogeneous media, singular coefficients]; Tyagi JPA(05) [Poisson, periodic boundary conditions]; Moroz JPA(06)mp [Helmholtz and Laplace, quasi-periodic]; Franklin a1202 [for Neumann boundary conditions].
@ For non-linear equations: Frasca MPLA(07)ht/07 [and quantum field theory applications]; Frasca IJMPA(08)-a0704 [short-time expansion]; Frasca & Khurshudyan IJMPC(18)-a1806 [higher-order non-linear equations].
> Online resources: see Wikipedia page.

For Classical Field Theory > s.a. gravitational radiation; huygens principle [tails].
* Interpretation: The Green function G(x, x') is the field produced at x by a unit-strength point source at a given point x'.
* In electrodynamics: It is used to write the electrostatic potential as

\[ \phi(x) = \int_V {\rm d}v'\,\rho(x')\,G(x,x') + {1\over4\pi}\oint_{\partial V} {\rm d}{\bf a}'\cdot(G\,\nabla'\!\phi-\phi\,\nabla'\!G)\;. \]

@ General references: Green 1828-a0807; in Morse & Feschbach 53; Barton 89; Cornwall et al 11 [gauge theories, pinch technique]; in Alastuey et al 16.
@ In curved spacetime: Waylen PRS(78) [early universe, singular and regular terms]; Molnár CQG(01)gq [electrostatic, in Schwarzschild spacetime]; Higuchi & Lee PRD(08)-a0807, Higuchi et al PRD(09) [retarded, in de Sitter space]; Esposito & Roychowdhury IJGMP(09) [spin-1/2 and 3/2, de Sitter space]; Chu & Starkman PRD(11)-a1108 [scalar, photon and graviton retarded Green's functions in perturbed spacetimes, perturbation theory]; Kazinski a1211 [stationary, slowly-varying spacetime]; Capoferri et al CAG-a1902 [on closed Riemannian manifolds]; Casals et al PRD(19)-a1910 [Schwarzschild spacetime, regularized calculation]; > s.a. klein-gordon fields in curved spacetime [Kerr spacetime].
@ Generalized: Xu & Yau JCTA(13) [Chung-Yau's discrete Green function]; Ray a1409 [exact Green functions on lattices]; Sanchez Sanchez & Vickers JPCS(18)-a1711 [Green operators on low-regularity spacetimes].

For Other Classical Systems > see Kadanoff-Baym Equations [transport].

For Quantum Systems > s.a. feynman propagator; green functions in quantum field theory; quantum oscillator.
@ References: Tsaur & Wang AJP(06)jul [Schrödinger equation]; Miyazawa JPA(06) [1D, in terms of reflection coefficients]; Brouder et al PRL(09) [many-body degenerate systems]; Baker a2008 [Lanczos recursion on a quantum computer, continued-fraction representation].

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