Green Function Equations

Formula

For a linear operator $L$ ,

L_xG(x,x')=\delta(x-x'), \qquad \psi(x)=\int dx'\,G(x,x')\eta(x')

solves $L\psi=\eta$ with the same boundary prescription. For a time-independent Hamiltonian,

G(z)=(z-H)^{-1}, \qquad (z-H)G(z)=I,

and, for a discrete spectrum,

G(z)=\sum_n\frac{|n\rangle\langle n|}{z-E_n}.

Boundary values are

G^\pm(E)=(E-H\pm i0)^{-1},

and the resolvent identity is

G(z)-G(w)=(w-z)G(z)G(w).

The operator, variables, measure, domain, and boundary prescription are part of the definition.
$G^+$ here uses $(E-H+i0)^{-1}$ ; other Fourier conventions can reverse signs and factors of $i$ .
Spectral formulas assume a self-adjoint $H$ and the appropriate sum-plus- integral spectral resolution.

Symbol	Meaning
$z$	complex spectral parameter off the spectrum
$E\pm i0$	boundary value from the upper or lower half-plane
$G(x,x';z)$	coordinate kernel of the resolvent
$\eta$	source term

There is no unique “Green function” without a prescription.
A resolvent is not the same object as the time-evolution kernel, although transforms relate them.
The $i0$ prescription is physical and analytic information, not notation that may be dropped.
Traces and density-of-states formulas may need finite-volume, per-volume, or subtracted definitions.
Numerical $i\eta$ broadening changes spectral resolution; convergence in both discretization and $\eta$ must be tested.

The inverse-kernel interpretation, spectral poles and residues, free-particle checks, exercises, and references are at What Is a Green Function?.

Detailed coordinate equations are at Energy Green Function.