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].

@ __Discrete__: Xu & Yau JCTA(13) [Chung-Yau's discrete Green function]; Ray a1409 [exact Green functions on lattices].

**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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