Green Functions

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

L G(x, x') = (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 = j, given the source j and the boundary conditions on the field , i.e., to propagate the field; It is thus also called propagator.
@ 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].
@ For non-linear equations: Frasca MPLA(07)ht/07 [and quantum field theory applications]; Frasca a0704 [short-time expansion].

For Classical Field Theory > s.a. gravitational radiation.
* Interpretation: The field produced by a unit strength point source at a given point x.
* In electrodynamics: Used to write the electrostatic potential as

(x) = _V dv' (x') G(x, x') + (1/4) _bdry(V) da' · (G ' – 'G) .

@ General references: Green 1828-a0807; in Morse & Feschbach 53; Barton 89.
@ In curved spacetime: Waylen PRS(78) [early universe, singular and regular terms]; Molnár CQG(01)gq [electrostatic, in Schwarzschild].

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

For Quantum Systems > s.a. feynman propagator; green functions in quantum field theory; quantum oscillator.
@ References: Tsaur & Wang AJP(06) [Schrödinger equation]; Miyazawa JPA(06) [1D, ito reflection coefficients].

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