Energy Green Function

The energy Green function is the coordinate-space kernel of a resolvent boundary value. With the outgoing convention,

$$G^+(x,x';E) = \left\langle x\left\rvert \frac{1}{E-H+i0} \right\lvert x'\right\rangle.$$

It answers an energy-domain source problem:

$$(E-H_x)G^+(x,x';E) = \delta(x-x'),$$

together with the spatial and asymptotic boundary conditions encoded by $H$ and $+i0$.

This page is the canonical home for the coordinate kernel, its differential equation, and the free outgoing examples. The abstract operator-valued inverse belongs to Resolvent Operator, while the general discrete-plus-continuum expansion belongs to Spectral Representation of Green Functions.

Definition

For a self-adjoint Hamiltonian $H$, the resolvent is

$$R(z)=(zI-H)^{-1}$$

when $z$ lies outside the spectrum. In a coordinate basis,

$$G(x,x';z) = \langle x\rvert R(z)\lvert x'\rangle.$$

For a real energy in the continuous spectrum, the inverse does not exist as an ordinary bounded operator at $z=E$. One instead takes a boundary value:

$$G^\pm(x,x';E) = \lim_{\epsilon\to0^+} \left\langle x\left\rvert \frac{1}{E-H\pm i\epsilon} \right\lvert x'\right\rangle.$$

The superscripts have physical content:

Kernel	Resolvent boundary value	Typical asymptotic condition
$G^+$	$(E-H+i0)^{-1}$	outgoing
$G^-$	$(E-H-i0)^{-1}$	incoming

For a time-reversal-invariant real Hamiltonian and real $E$ where both boundary values are defined,

$$G^-(x,x';E) = G^+(x',x;E)^*.$$

The energy Green function is not a time-evolution kernel. It has dimensions of inverse energy times the coordinate delta distribution. In $d$ Cartesian dimensions,

$$[G]=\frac{1}{\text{energy}\,\text{length}^d}.$$

Differential Equation

The operator identity

$$(zI-H)R(z)=I$$

implies, after taking coordinate matrix elements,

$$(z-H_x)G(x,x';z) = \delta(x-x').$$

For a one-dimensional Schrödinger Hamiltonian

$$H = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} +V(x),$$

the Green function satisfies

$$\left[ E-V(x) +\frac{\hbar^2}{2m}\frac{d^2}{dx^2} \right] G^+(x,x';E) = \delta(x-x').$$

Away from $x=x'$, it solves the homogeneous stationary Schrödinger equation. At the source point, $G$ is continuous for an ordinary finite potential, while its first derivative has a fixed jump. Integrating across $x'$ gives

$$\left. \frac{\partial G}{\partial x} \right|_{x=x'^+} - \left. \frac{\partial G}{\partial x} \right|_{x=x'^-} = \frac{2m}{\hbar^2}.$$

This jump condition fixes the overall normalization after the left and right homogeneous solutions have been chosen.

Response to an energy-domain source

If

$$(E-H)\psi(x)=f(x),$$

then a solution with the boundary prescription carried by $G$ is

$$\psi(x) = \int dx'\, G(x,x';E)f(x').$$

The integration measure and domain must match the coordinate basis. On a curved or radial configuration space, the delta function and integral carry the corresponding measure factors.

Boundary Conditions

An inverse differential equation does not select a unique Green function until boundary conditions are supplied. The energy Green function must satisfy:

• the operator-domain conditions of $H$ in the coordinate variable $x$;
• the adjoint conditions in $x'$;
• the source normalization at $x=x'$;
• an asymptotic prescription for continuum energies.

For a hard-wall interval, $G$ vanishes at each Dirichlet endpoint. For a radial problem, regularity at the origin and the radial measure matter. For a bound-state problem below threshold, the kernel decays at spatial infinity. For positive-energy scattering on the full line or in three dimensions, $G^+$ selects outgoing waves and $G^-$ selects incoming waves.

In three dimensions, the outgoing Sommerfeld condition has the schematic form

$$\lim_{r\to\infty} r\left( \frac{\partial}{\partial r}-ik \right) G^+(\mathbf r,\mathbf r';E) =0,$$

with $\mathbf r'$ fixed and

$$E=\frac{\hbar^2k^2}{2m}.$$

The incoming kernel uses the opposite sign. Spatial boundaries and the $i0$ prescription solve different parts of the specification: the former define the Hamiltonian domain, while the latter select a continuum boundary value at infinity.

The iε Prescription

The notation

$$\frac{1}{E-H\pm i0}$$

means

$$\lim_{\epsilon\to0^+} \frac{1}{E-H\pm i\epsilon}.$$

The scalar distribution identity

$$\frac{1}{x\pm i0} = \operatorname{PV}\frac{1}{x} \mp i\pi\delta(x)$$

lifts through the spectral theorem to

$$R^+(E)-R^-(E) = -2\pi i\,\delta(E-H).$$

Consequently,

$$-\frac{1}{\pi} \operatorname{Im}G^+(x,x;E)$$

is the local spectral density under the stated convention. The trace gives the total density of states when that trace is well defined or appropriately regularized. Those formulas are developed in Green Functions and Density of States.

For finite $\epsilon=\eta\gt0$,

$$G_\eta(E) = \frac{1}{E-H+i\eta}$$

is a smoothened resolvent. Numerically, $\eta$ broadens delta-function spectral lines into Lorentzian peaks. It is useful, but it is not identical to the exact $i0$ limit.

Relation to retarded time evolution

For a time-independent Hamiltonian,

$$\begin{aligned} \frac{1}{E-H+i0} &= -\frac{i}{\hbar} \int_0^\infty dt\, e^{i(E+i0)t/\hbar} e^{-iHt/\hbar}. \end{aligned}$$

Thus $G^+(E)$ is the energy transform of a retarded time-domain evolution kernel in this convention. The step-function normalization and advanced counterpart belong to Retarded and Advanced Green Functions.

Free Particle in One Dimension

Let

$$H_0 = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2}, \qquad E=\frac{\hbar^2k^2}{2m}, \qquad k\gt0.$$

The outgoing Green function solves

$$\left( \frac{d^2}{dx^2}+k^2 \right) G_0^+(x,x';E) = \frac{2m}{\hbar^2}\delta(x-x').$$

For $x\ne x'$, outgoing behavior requires a wave traveling away from the source point on each side. The solution is

$$G_0^+(x,x';E) = -\frac{im}{\hbar^2k} e^{ik\lvert x-x'\rvert}.$$

Its derivative jump is

$$\begin{aligned} &\left. \partial_xG_0^+ \right|_{x=x'^+} - \left. \partial_xG_0^+ \right|_{x=x'^-}\\ &\qquad= \frac{2m}{\hbar^2}, \end{aligned}$$

which verifies the delta-function normalization. The incoming kernel is the complex conjugate:

$$G_0^-(x,x';E) = \frac{im}{\hbar^2k} e^{-ik\lvert x-x'\rvert}.$$

The absolute-value displacement makes the physical meaning transparent: for $x\gt x'$, $G_0^+$ carries $e^{ikx}$ to the right; for $x\lt x'$, it carries $e^{-ikx}$ to the left. Both pieces move away from the source.

Free Particle in Three Dimensions

The momentum representation is

$$\begin{aligned} G_0^+(\mathbf r,\mathbf r';E) &= \int\frac{d^3p}{(2\pi\hbar)^3} \frac{e^{i\mathbf p\cdot(\mathbf r-\mathbf r')/\hbar}} {E-\mathbf p^2/(2m)+i0}. \end{aligned}$$

Let

$$R=\lvert\mathbf r-\mathbf r'\rvert, \qquad k=\frac{\sqrt{2mE}}{\hbar}.$$

For $E\gt0$, contour integration or the outgoing Helmholtz fundamental solution gives

$$G_0^+(\mathbf r,\mathbf r';E) = -\frac{m}{2\pi\hbar^2} \frac{e^{ikR}}{R}.$$

The distributional identity

$$(\nabla^2+k^2) \frac{e^{ikR}}{R} = -4\pi\delta^{(3)} (\mathbf r-\mathbf r')$$

checks the normalization. At large $r$ with $\mathbf r'$ fixed,

$$G_0^+ \sim \frac{e^{ikr}}{r} \times \text{an angular phase},$$

so $+i0$ has selected an outgoing spherical wave.

For $E\lt0$, define

$$\kappa=\frac{\sqrt{-2mE}}{\hbar}.$$

Then the decaying free resolvent is

$$G_0(\mathbf r,\mathbf r';E) = -\frac{m}{2\pi\hbar^2} \frac{e^{-\kappa R}}{R}.$$

Below the continuum threshold there is no outgoing-versus-incoming oscillatory ambiguity. The physically appropriate spatial condition is decay.

Bound-State Poles

Suppose $H$ has normalized isolated bound states

$$H\lvert n\rangle=E_n\lvert n\rangle.$$

Their contribution to the coordinate Green function is

$$G(x,x';z) \supset \sum_n \frac{\psi_n(x)\psi_n^*(x')}{z-E_n}.$$

Near a nondegenerate isolated eigenvalue,

$$G(x,x';z) = \frac{\psi_n(x)\psi_n^*(x')}{z-E_n} + G_{\mathrm{regular}}(x,x';z).$$

Therefore

$$\operatorname*{Res}_{z=E_n} G(x,x';z) = \psi_n(x)\psi_n^*(x').$$

For a degenerate eigenspace, the residue is the full projector kernel

$$\langle x\rvert P_n\lvert x'\rangle.$$

Poles reveal bound-state energies, while residues reveal their wavefunction or projector content. Continuum thresholds instead produce boundary-value discontinuities and branch-cut structure. Resonance poles require analytic continuation away from the physical sheet and should not be confused with normalizable bound-state poles.

The full organization belongs to Spectral Representation of Green Functions and Bound States and Scattering Poles.

Scattering Preview

Let

$$H=H_0+V,$$

and consider a free incoming state $\lvert\mathbf k\rangle$ at energy $E$. The outgoing Lippmann–Schwinger equation is

$$\lvert\psi_{\mathbf k}^{(+)}\rangle = \lvert\mathbf k\rangle + \frac{1}{E-H_0+i0} V \lvert\psi_{\mathbf k}^{(+)}\rangle.$$

In coordinate space,

$$\begin{aligned} \psi_{\mathbf k}^{(+)}(\mathbf r) &= e^{i\mathbf k\cdot\mathbf r} + \int d^3r'\, G_0^+(\mathbf r,\mathbf r';E)\\ &\qquad\times V(\mathbf r') \psi_{\mathbf k}^{(+)}(\mathbf r'). \end{aligned}$$

The factor $e^{ik\lvert\mathbf r-\mathbf r'\rvert}/\lvert\mathbf r-\mathbf r'\rvert$ produces the outgoing spherical part of the asymptotic wave. Replacing $+i0$ by $-i0$ selects incoming asymptotics instead.

This page owns the coordinate-kernel definition, source normalization, and free examples. Green Function for Scattering owns the scattering-specific passage from a continuum boundary value through far-field factorization to the amplitude. Lippmann–Schwinger Equation Preview develops the operator bridge from resolvents to formal scattering states, while Lippmann–Schwinger Equation owns the exact state equation.

Common Mistakes

• Calling $(E-H)^{-1}$ well defined on the continuous spectrum without a boundary value or regularization.
• Omitting the coordinate delta function in the defining differential equation.
• Solving the homogeneous equation on either side of the source but missing the derivative jump.
• Treating $+i0$ as decorative notation rather than the outgoing boundary prescription.
• Reversing the outgoing and incoming signs after changing Fourier conventions.
• Applying a full-line or free-space Green function when walls, topology, or radial domains require different spatial conditions.
• Confusing the energy Green function with the time-domain propagator kernel.
• Squaring $G$ and interpreting it as a normalized transition probability.
• Reading a finite numerical broadening $\eta$ as the exact infinitesimal limit.
• Treating every singularity as a normalizable bound state and ignoring thresholds, cuts, and resonances.
• Dropping degenerate-state projectors from pole residues.
• Using the outgoing free Green function without matching the plane-wave normalization in a scattering calculation.

Cross-Links

• What Is a Green Function?
• Resolvent Operator
• Retarded and Advanced Green Functions
• Spectral Representation of Green Functions
• Green Functions and Density of States
• Free-Particle Propagator
• Propagators and Boundary Conditions
• Green Function Table
• Lippmann–Schwinger Equation Preview
• Lippmann–Schwinger Equation
• Green Function for Scattering
• Bound States and Scattering Poles

References

• E. N. Economou, Green’s Functions in Quantum Physics, 3rd ed., Springer, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Wiley, 1972.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics III: Scattering Theory, Academic Press, 1979.
• A. L. Fetter and J. D. Walecka, Quantum Theory of Many-Particle Systems, Dover, 2003.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Derive the differential equation for $G(x,x';z)$ from the resolvent identity.

Solution

Start with

$$(zI-H)R(z)=I.$$

Insert coordinate states:

$$\langle x\rvert (zI-H)R(z) \lvert x'\rangle = \langle x\vert x'\rangle.$$

The operator $H$ acts on the $x$ variable, so

$$(z-H_x) \langle x\rvert R(z)\lvert x'\rangle = \delta(x-x').$$

Using the definition of the coordinate kernel gives

$$(z-H_x)G(x,x';z) = \delta(x-x').$$

2. Determine the coefficient of the outgoing one-dimensional free Green function from its derivative jump.

Solution

Use the outgoing ansatz

$$G_0^+(x,x';E) = A e^{ik\lvert x-x'\rvert}.$$

For $x\gt x'$,

$$\partial_xG_0^+=ikA e^{ik(x-x')},$$

while for $x\lt x'$,

$$\partial_xG_0^+=-ikA e^{ik(x'-x)}.$$

At $x=x'$, the jump is

$$2ikA.$$

The source equation requires

$$2ikA=\frac{2m}{\hbar^2},$$

so

$$A = \frac{m}{i\hbar^2k} = -\frac{im}{\hbar^2k}.$$

3. Verify the normalization of the three-dimensional free outgoing Green function.

Solution

For

$$G_0^+(R;E) = -\frac{m}{2\pi\hbar^2} \frac{e^{ikR}}{R},$$

use

$$(\nabla^2+k^2) \frac{e^{ikR}}{R} = -4\pi\delta^{(3)}(\mathbf R).$$

Because

$$E-H_0 = \frac{\hbar^2}{2m} (\nabla^2+k^2),$$

one obtains

$$\begin{aligned} (E-H_0)G_0^+ &= \frac{\hbar^2}{2m} \left( -\frac{m}{2\pi\hbar^2} \right) (-4\pi)\delta^{(3)}\\ &= \delta^{(3)}. \end{aligned}$$

4. Show that the residue at a nondegenerate bound-state pole is the coordinate projector kernel.

Solution

Near $z=E_n$, separate the singular spectral term:

$$G(x,x';z) = \frac{\psi_n(x)\psi_n^*(x')}{z-E_n} + G_{\mathrm{regular}}(x,x';z).$$

Multiplying by $z-E_n$ and taking the limit gives

$$\begin{aligned} \lim_{z\to E_n} (z-E_n)G(x,x';z) &= \psi_n(x)\psi_n^*(x')\\ &= \langle x\rvert \lvert n\rangle\langle n\rvert \lvert x'\rangle. \end{aligned}$$

For degeneracy, sum over an orthonormal basis of the eigenspace to obtain $\langle x\rvert P_n\lvert x'\rangle$.

5. Use the distribution identity to derive the discontinuity between outgoing and incoming resolvents.

Solution

For a scalar variable,

$$\frac{1}{x+i0} - \frac{1}{x-i0} = -2\pi i\delta(x).$$

Apply this identity to each spectral value of $H$ through functional calculus:

$$\begin{aligned} R^+(E)-R^-(E) &= \frac{1}{E-H+i0} - \frac{1}{E-H-i0}\\ &= -2\pi i\delta(E-H). \end{aligned}$$

Taking coordinate matrix elements gives the corresponding discontinuity of $G^+(x,x';E)$ and $G^-(x,x';E)$.