Green's Function

The defining equation

Let L be a linear differential operator on functions of x (one or more variables). A Green's function for L is a kernel G(x, x′) satisfying

L_x G(x, x′) = δ(x − x′)

The subscript on L_x reminds you the operator acts on x, with x′ a parameter. The right-hand side δ(x − x′) is the Dirac delta — an idealized point source at location x′. G(x, x′) is the field response that L produces from this point source, subject to whatever boundary conditions are appropriate for the problem.

The key consequence: the inhomogeneous equation L u = f has the explicit solution

u(x) = ∫ G(x, x′) f(x′) dx′

Convolution against G turns any source f into the corresponding response u.

Why convolution against G works

Step through the logic carefully. Write the source as a continuous superposition of point sources:

f(x) = ∫ f(x′) δ(x − x′) dx′

The sifting property of δ makes the integral on the right equal f(x) trivially. Now the response to a point source at x′ is, by definition of G, the function G(x, x′). Linearity of L means the response to a superposition is the superposition of responses:

u(x) = ∫ f(x′) G(x, x′) dx′

Verify by acting on both sides with L:

L_x u(x) = ∫ f(x′) L_x G(x, x′) dx′
        = ∫ f(x′) δ(x − x′) dx′
        = f(x)    ✓

Three steps — pull L inside the integral (because L acts only on x), substitute L G = δ, then sift. The Green's function method reduces solving any inhomogeneous problem to a single integral, provided you can find G once.

Example — Green's function for the 3D Laplacian

Take L = ∇² on ℝ³ with G → 0 at infinity. We need ∇² G(x, x′) = δ(x − x′).

By translation invariance, G depends only on r = |x − x′|. Spherical symmetry then gives ∇² G = (1/r²) d(r² dG/dr)/dr = 0 away from r = 0. Integrating: G = −A/r + B; the boundary condition G → 0 forces B = 0. To find A, integrate ∇² G = δ over a small ball of radius r centred at x′:

∫∫∫_{|x − x′| < r} ∇² G dV  =  1
∮_{∂ball} ∇G · n̂ dA          =  1    (divergence theorem)
(A / r²)(4π r²)               =  1
A                              =  1/(4π)

So the free-space Green's function of the 3D Laplacian is

G(x, x′) = −1 / (4π |x − x′|)

This is the Coulomb potential of a unit point source, up to a sign. The Poisson equation ∇²φ = −ρ/ε₀ has the closed-form solution

φ(x) = (1/(4πε₀)) ∫ ρ(x′) / |x − x′| dx′

— Coulomb's law generalized to a continuous charge distribution. The 1/r is not an accident of electrostatics; it is the Green's function of the Laplacian in three dimensions.

Example — Heat kernel

For the diffusion operator L = ∂/∂t − D ∇² on ℝⁿ × ℝ, the Green's function is the famous heat kernel:

G(x, t; x′, t′) = Θ(t − t′) · (4πD(t − t′))^{−n/2} · exp(−|x − x′|² / 4D(t − t′))

Θ is the unit step (causality — nothing diffuses backward in time). A point source of heat deposited at x′ at time t′ spreads as a Gaussian whose variance grows linearly: σ² = 2D(t − t′). The diffusion constant D sets the rate. Convolving any initial temperature profile u(x, 0) against G(x, t; x′, 0) propagates it forward to time t.

Example — Wave equation, retarded Green's function

For L = ∂²/∂t² − c²∇² (the d'Alembertian on ℝ³), the retarded Green's function is

G_R(x, t; x′, t′) = δ(t − t′ − |x − x′|/c) / (4π |x − x′|)

The δ function enforces a hard retardation — the field at (x, t) only "knows" about the source at the single earlier time t′ = t − |x − x′|/c, when a light-speed signal could have left x′ and arrived at x. Effects come after sources, never before. This is the source of retarded potentials in electromagnetism — the Liénard–Wiechert potential is the convolution of a charge–current source with this G_R.

Boundary conditions and image charges

The PDE L G = δ has many solutions; they differ by homogeneous solutions of L u = 0. Boundary conditions pick out the unique G appropriate to the problem.

Method of images: to solve ∇²φ = −ρ/ε₀ in the half-space z > 0 with φ = 0 on z = 0 (a grounded conducting plane), use

G(x, x′) = −1/(4π) · [1/|x − x′| − 1/|x − x′_image|]

where x′_image is the mirror image of x′ across z = 0. The second term cancels G on the plane, satisfying the boundary condition. For more complex geometries, conformal maps (2D) or eigenfunction expansions are used to construct G.

Eigenfunction expansion of G

When L is self-adjoint with eigenfunctions L φ_n = λ_n φ_n satisfying the same boundary conditions, the Green's function has the spectral representation

G(x, x′) = ∑_n  φ_n(x) φ_n*(x′) / λ_n

(Assuming λ_n ≠ 0 for all n; otherwise modifications are needed.) This is the inverse of L expressed in its eigenbasis — divide each eigenvalue by 1 over itself, exactly as one inverts a diagonal matrix. For continuous spectra, the sum becomes an integral. Spectral Green's functions are the foundation of scattering theory, normal-mode analysis, and quantum perturbation theory.

Green's functions across operators

Operator L	Where it lives	Free-space G	Physical meaning
∇² (3D Laplacian)	ℝ³	−1 / (4π r)	Coulomb / Newton potential
∇² (2D)	ℝ²	(1 / 2π) ln r	2D electrostatics, vortex
d²/dx² (1D)	ℝ	−|x − x′| / 2	1D string under point load
∂/∂t − D∇²	ℝⁿ × ℝ	(4πDt)^{−n/2} e^{−r²/4Dt}	Heat kernel / diffusion
∂²/∂t² − c²∇²	ℝ³ × ℝ	δ(t − r/c) / (4π r)	Retarded EM potential
−∇² + m² (Yukawa)	ℝ³	e^{−mr} / (4π r)	Screened Coulomb / massive boson
iℏ ∂/∂t − Ĥ	ℝⁿ × ℝ	Feynman propagator K(x, t; x′, t′)	QM probability amplitude

The same idea — invert L by convolving against its point-source response — produces every entry. The form of G differs because the operator, dimension, and boundary conditions differ; the conceptual move is identical.

Reciprocity — why G is symmetric

For a self-adjoint operator L = L† (with L's domain making it self-adjoint via the relevant boundary conditions), Green's identity gives

G(x, x′) = G(x′, x)

Physically: the field at point x produced by a unit source at x′ equals the field at x′ produced by a unit source at x. This is reciprocity — antennas show it (transmit-receive symmetry), so do passive optical components and elastic structures.

For non-self-adjoint operators (e.g., diffusion with convection ∂_t + v·∇ − D∇²), reciprocity fails. You need the Green's function G and the adjoint Green's function G†, where G† satisfies L† G† = δ.

Where Green's functions appear

• Electrostatics and gravity. The 1/r Newton/Coulomb potential is the Green's function of the Laplacian. Multipole expansions are eigenfunction expansions of G in spherical harmonics.
• Heat conduction and finance. The heat kernel is the Green's function of the diffusion equation. Black–Scholes option pricing reduces to a backward heat equation; its Green's function gives the option-price formula directly.
• Electromagnetic radiation. Retarded Green's function of the d'Alembertian gives Liénard–Wiechert potentials — the EM field of an arbitrarily moving charge.
• Solid-state physics. Green's function methods compute electronic spectra, density of states, and transport properties. Many-body Green's functions handle interactions perturbatively.
• Quantum field theory. Feynman propagators G_F are Green's functions of the kinetic operator with an iε prescription. Feynman diagrams are graphical convolutions of propagators with interaction vertices.
• Scattering theory. The Lippmann–Schwinger equation ψ = φ + G V ψ rearranges Schrödinger's equation as a fixed-point problem in terms of the free Green's function G.
• Stochastic processes. The transition density of a diffusion process is exactly the Green's function of the Fokker–Planck operator (forward) or its adjoint (backward Kolmogorov).

Common mistakes

• Forgetting boundary conditions. The Green's function depends critically on the boundary. The 3D Laplacian's free-space G is −1/(4π r); on a half-space with Dirichlet boundary, it has an image term; on a sphere, an entirely different form. Use the right G for the problem.
• Treating ∂_t G as ordinary derivative at t = t′. Causal Green's functions have a step discontinuity at t = t′; differentiating them produces δ functions that show up correctly in L G = δ. Symbolic computation tools sometimes need the discontinuity built in explicitly.
• Confusing retarded and advanced. For physical electromagnetism use G_R (cause precedes effect). G_A appears in time-reversal arguments; using G_A in a forward-causal problem gives unphysical answers.
• Assuming reciprocity always holds. Only for self-adjoint operators. Diffusion–convection and dissipative systems have G(x, x′) ≠ G(x′, x); two distinct Green's functions are needed.
• Singularities at x = x′. G is typically singular as x → x′ (1/r in 3D, log r in 2D). When evaluating u(x) = ∫ G f, special care is needed at the diagonal — principal-value integrals or distributional interpretations.