Green Functions| 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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