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(*x*–*x'*) ;

Notice that a given operator *L* has many Green functions, depending
on the boundary conditions imposed on the solution.

* __Applications__: Used
to find solutions of the de *L**φ* = *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 / |*x* – *x'*|; 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].

> __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

*φ*(*x*)
= ∫_{V}
d*v'* *ρ*(*x'*) *G*(*x*,* x'*)
+ (1/4π) ∫_{}_{∂(V)} d**a***'* · (*G* ∇*'**φ* – *φ* ∇*'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];
> 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].

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