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__: It is
used to find solutions of the differential equation *Lφ*
= *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];
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

@ __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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