Lippmann–Schwinger Equation Preview

The Lippmann–Schwinger equation turns a stationary scattering problem into a free incident state plus a wave emitted by an effective source. For

$$H=H_0+V,$$

an outgoing scattering state at energy $E$ is written formally as

$$\lvert\psi^{(+)}\rangle = \lvert\phi\rangle + \frac{1}{E-H_0+i0} V\lvert\psi^{(+)}\rangle.$$

The three terms have distinct roles:

• $\lvert\phi\rangle$ is a prescribed free solution at energy $E$;
• $V\lvert\psi^{(+)}\rangle$ acts as the source generated inside the interaction region;
• $(E-H_0+i0)^{-1}$ propagates that source with outgoing boundary conditions.

This page is the canonical operator-level bridge between resolvents and scattering states. The exact coordinate state equation belongs to Lippmann–Schwinger Equation, while repeated iteration and convergence belong to Born Series in the scattering volume.

Free and Interacting Hamiltonians

Split the Hamiltonian into a reference problem and an interaction:

$$H=H_0+V.$$

Usually $H_0$ describes free relative motion and $V$ is a sufficiently short-range potential. A stationary interacting state obeys

$$(E-H)\lvert\psi\rangle=0,$$

or equivalently

$$(E-H_0)\lvert\psi\rangle = V\lvert\psi\rangle.$$

This has the form of an inhomogeneous free equation. The quantity $V\lvert\psi\rangle$ is not an externally prescribed source because it contains the unknown state, but the Green-function logic is the same: invert the free operator with a boundary prescription and impose self-consistency.

The incident state satisfies the homogeneous free equation

$$(E-H_0)\lvert\phi\rangle=0.$$

It cannot be discarded. At a continuum energy, $E-H_0$ is not ordinarily invertible, and the general solution of the inhomogeneous equation equals one particular solution plus a homogeneous solution. In scattering, $\lvert\phi\rangle$ specifies the beam or channel incident on the interaction region.

The choice of $H_0$ is part of the formulation. For an unscreened Coulomb force, ordinary plane waves and free spherical asymptotics are not the correct long-distance reference; Coulomb Scattering treats that long-range modification.

Resolvent Form

For complex $z$ outside the spectra, define the free and full resolvents

$$R_0(z)=(z-H_0)^{-1}, \qquad R(z)=(z-H)^{-1}.$$

Since

$$z-H = (z-H_0)-V,$$

the second resolvent identity gives

$$R(z) = R_0(z) + R_0(z)VR(z).$$

An equally valid ordering is

$$R(z) = R_0(z) + R(z)VR_0(z).$$

These are exact operator identities wherever the resolvents exist. Iteration produces the formal resolvent series

$$R = R_0 + R_0VR_0 + R_0VR_0VR_0 + \cdots,$$

but convergence requires more than writing the series. The Resolvent Operator page owns the abstract inverse and analytic identities; the scattering volume owns the approximation theory built from iteration.

At a real continuum energy, take boundary values

$$R_0^{(\pm)}(E) = \lim_{\epsilon\to0^+} \frac{1}{E-H_0\pm i\epsilon}.$$

The stationary equation then becomes the Lippmann–Schwinger equation

$$\lvert\psi^{(\pm)}\rangle = \lvert\phi\rangle + R_0^{(\pm)}(E) V\lvert\psi^{(\pm)}\rangle.$$

Applying $E-H_0$ recovers

$$(E-H_0)\lvert\psi^{(\pm)}\rangle = V\lvert\psi^{(\pm)}\rangle$$

in the appropriate generalized sense. The homogeneous incident state is precisely what would be lost by treating $R_0(E)$ as an ordinary inverse and multiplying naively.

Incoming and Outgoing Boundary Conditions

For the time convention $e^{-iEt/\hbar}$, the free three-dimensional coordinate kernels at

$$E=\frac{\hbar^2k^2}{2m}, \qquad k\gt0,$$

are

$$G_0^{(\pm)}(\mathbf r,\mathbf r';E) = -\frac{m}{2\pi\hbar^2} \frac{ e^{\pm ik\lvert\mathbf r-\mathbf r'\rvert} }{ \lvert\mathbf r-\mathbf r'\rvert }.$$

Thus $+i0$ selects a wave moving outward from a localized source, while $-i0$ selects the incoming counterpart. The normalization and source equation are derived in Energy Green Function.

The notation is easy to misread:

State	Free component	Scattered boundary condition
$\lvert\psi^{(+)}\rangle$	prescribed incident state	outgoing from the interaction region
$\lvert\psi^{(-)}\rangle$	prescribed asymptotic state	incoming toward the interaction region

The superscript does not label positive or negative energy. Nor does it mean that the whole $\lvert\psi^{(+)}\rangle$ state is outgoing: it contains an incident free component and an outgoing scattered component.

Energy-domain outgoing behavior and time-domain retardation are related by analytic boundary values, but they answer different formulations of a problem. In the stationary scattering equation, the immediate role of $i0$ is to select the spatial radiation condition at infinity.

Formal Scattering States

Continuum scattering states are generally not normalizable Hilbert-space vectors. They are generalized eigenstates with delta-function normalization, or distributions in a rigged-Hilbert-space description. Accordingly, the limit

$$R_0(E\pm i0)$$

may exist only between suitable weighted spaces or inside matrix elements. The notation is compact, but it encodes a limiting-absorption statement.

When wave operators exist, one may write

$$\lvert\psi^{(\pm)}\rangle = \Omega^{(\pm)}\lvert\phi\rangle,$$

with

$$\Omega^{(\pm)} = \underset{t\to\mp\infty}{\operatorname{s-lim}} e^{iHt/\hbar}e^{-iH_0t/\hbar}.$$

The $+$ state is matched to free data in the distant past and develops outgoing scattered radiation; the $-$ state is matched in the distant future. Existence, isometry, and asymptotic completeness require hypotheses on $V$. They are not automatic consequences of writing the formal limit.

Bound states lie outside the scattering subspace generated by these wave operators. Resonances are also not ordinary normalizable scattering eigenstates; they appear through analytic continuation. Their pole structure is discussed in Bound States and Scattering Poles.

Link to Scattering Theory

Taking a coordinate matrix element gives

$$\begin{aligned} \psi^{(+)}(\mathbf r) &= \phi(\mathbf r) + \int d^3r'\, G_0^{(+)}(\mathbf r,\mathbf r';E)\\ &\qquad\times V(\mathbf r')\psi^{(+)}(\mathbf r'). \end{aligned}$$

This makes the source picture concrete: $V(\mathbf r')\psi^{(+)}(\mathbf r')$ emits an outgoing wave from every point in the interaction region, and the exact state is the self-consistent sum of those contributions plus the incident wave.

The next steps have separate canonical homes:

Question	Canonical page
How is the large-$r$ coefficient extracted?	Scattering Amplitude
How is the integral equation derived and iterated?	Lippmann–Schwinger Equation
When may the exact state under the integral be replaced by the incident state?	First Born Approximation
How does the amplitude become an observable rate?	Cross Sections
How are rotationally invariant problems organized?	Partial-Wave Expansion

Common Mistakes

• Treating $(E-H_0)^{-1}$ as an ordinary bounded inverse at a continuum energy.
• Multiplying by a resolvent and accidentally dropping the incident homogeneous solution.
• Reading the superscripts $(\pm)$ as signs of the energy.
• Calling $\lvert\psi^{(+)}\rangle$ a purely outgoing state and forgetting its incident component.
• Replacing $+i0$ with a finite absorption without stating that the physical problem has changed.
• Iterating the equation and assuming the resulting Born series converges for every potential and energy.
• Applying free short-range asymptotics to an unscreened Coulomb potential.
• Treating plane waves as normalized vectors rather than generalized states.
• Assuming formal wave operators exist and are complete without conditions on the interaction.
• Confusing bound-state poles, resonance poles, and continuum boundary values.

Cross-Links

• Resolvent Operator
• Energy Green Function
• Lippmann–Schwinger Equation
• Scattering Amplitude
• First Born Approximation
• Cross Sections
• Partial-Wave Expansion
• Coulomb Scattering
• Bound States and Scattering Poles

References

• J. R. Taylor, Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover, 2006.
• R. G. Newton, Scattering Theory of Waves and Particles, 2nd ed., Springer, 1982.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics III: Scattering Theory, Academic Press, 1979.
• D. R. Yafaev, Mathematical Scattering Theory: General Theory, American Mathematical Society, 1992.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.

Exercises

1. Derive $R=R_0+R_0VR$ from the definitions of the free and full resolvents.

Solution

Use

$$R^{-1}=z-H_0-V, \qquad R_0^{-1}=z-H_0.$$

Then

$$R_0^{-1}-R^{-1}=V.$$

Multiplying on the left by $R_0$ and on the right by $R$ gives

$$R-R_0=R_0VR.$$

Therefore

$$R=R_0+R_0VR.$$

Reversing the multiplication order gives the companion identity $R=R_0+RVR_0$.

2. Verify formally that a solution of the Lippmann–Schwinger equation satisfies the stationary Schrödinger equation.

Solution

Start from

$$\lvert\psi^{(\pm)}\rangle = \lvert\phi\rangle + R_0^{(\pm)}V \lvert\psi^{(\pm)}\rangle.$$

Apply $E-H_0$. Since $(E-H_0)\lvert\phi\rangle=0$ and, in the generalized boundary-value sense,

$$(E-H_0)R_0^{(\pm)}=I,$$

one obtains

$$(E-H_0)\lvert\psi^{(\pm)}\rangle = V\lvert\psi^{(\pm)}\rangle.$$

Moving the right side to the left gives $(E-H)\lvert\psi^{(\pm)}\rangle=0$.

3. Why must the incident state be added even though a Green function is called an inverse?

Solution

At a continuum energy, $E-H_0$ has nontrivial generalized homogeneous solutions and no ordinary bounded inverse. A Green-function boundary value supplies one particular solution of

$$(E-H_0)\lvert\psi\rangle = V\lvert\psi\rangle,$$

but the general solution also contains a homogeneous term. The chosen $\lvert\phi\rangle$ specifies the incoming channel. Omitting it would erase the incident beam and change the boundary-value problem.

4. For the convention $e^{-iEt/\hbar}$, identify the spatial behavior selected by $R_0(E+i0)$ in three dimensions.

Solution

Its coordinate kernel is proportional to

$$\frac{e^{ik\lvert\mathbf r-\mathbf r'\rvert}} {\lvert\mathbf r-\mathbf r'\rvert}.$$

At large distance from a localized source, this behaves as $e^{ikr}/r$, an outgoing spherical wave for the stated time convention. Thus $E+i0$ selects outgoing scattered radiation. The opposite boundary value produces $e^{-ikr}/r$ and incoming behavior.

5. What is the first approximation produced by iterating the Lippmann–Schwinger equation once, and why can it fail near a resonance?

Solution

Insert the incident state for the exact state on the right:

$$\lvert\psi^{(+)}\rangle \approx \lvert\phi\rangle + R_0^{(+)}V\lvert\phi\rangle.$$

This is the first Born approximation at the level of the state. Near a resonance, repeated propagation and interaction can be enhanced by small energy denominators. The omitted terms $R_0VR_0V\lvert\phi\rangle$, and higher iterations, need not be small even when a naive potential-strength estimate looks modest.