Lippmann–Schwinger Numerics

The Lippmann–Schwinger equation computes scattering through an integral kernel instead of outward radial propagation. Its numerical difficulty is the free resolvent’s on-shell singularity. Replacing that singularity by an arbitrary small imaginary number changes both quadrature resolution and the physical limiting procedure.

Required background. The Lippmann–Schwinger equation defines the outgoing resolvent; numerical error and stability explains conditioning and the interpretation of residuals.

The radial momentum integral

Use wave-number variables $p,q,k$ and $\alpha=\hbar^2/(2\mu)$, so $E=\alpha k^2$. Choose the reduced partial-wave kernel convention

$$T(p,k)=V(p,k)+ \int_0^\Lambda dq\, \frac{q^2V(p,q)T(q,k)}{E-\alpha q^2+i0}.$$

All angular normalization constants are absorbed into the defined kernels $V,T$; conversion to a physical amplitude must use this same convention. $\Lambda$ is a momentum cutoff and must be varied unless it is part of the declared model.

For $0<k<\Lambda$, the distribution identity separates the integral into a principal value and an on-shell term:

$$\int_0^\Lambda\frac{q^2F(q)\,dq}{E-\alpha q^2+i0} = \frac1\alpha\operatorname{PV}\int_0^\Lambda \frac{q^2F(q)}{k^2-q^2}\,dq -\frac{i\pi k}{2\alpha}F(k).$$

The negative imaginary sign encodes the outgoing prescription.

Pole subtraction before quadrature

For sufficiently smooth $F$, rewrite the principal value as

$$\operatorname{PV}\int_0^\Lambda \frac{q^2F(q)}{k^2-q^2}\,dq =\int_0^\Lambda \frac{q^2F(q)-k^2F(k)}{k^2-q^2}\,dq +\frac{kF(k)}2 \log\left|\frac{\Lambda+k}{\Lambda-k}\right|.$$

The first integrand has a removable singularity. Its limiting value at $q=k$ is $-F(k)-kF'(k)/2$. Use that limit or an appropriate interpolant when a quadrature node lands on shell.

The on-shell value $T(k,k)$ is an additional unknown unless the grid includes $k$. A Nyström discretization can include it explicitly, producing a linear system $(I-K)t=v$. Solve the system; do not explicitly form its inverse. Monitor the residual and condition estimate, especially near a pole.

A separable-kernel benchmark

Let $V(p,q)=\lambda g(p)g(q)$ with real $\lambda$ and real $g$, so the potential kernel is Hermitian. Then

$$T(p,k)=g(p)\tau(E)g(k), \qquad \tau(E)=\frac{\lambda}{1-\lambda I(E)},$$

where $I(E)=\int_0^\Lambda q^2g(q)^2/(E-\alpha q^2+i0)\,dq$. For the cutoff model $g=1$,

$$I(E)=\frac1\alpha \left[-\Lambda+\frac k2 \log\left|\frac{\Lambda+k}{\Lambda-k}\right| -\frac{i\pi k}{2}\right].$$

This tests both the subtracted real integral and the imaginary on-shell contribution. The cutoff model is not automatically a cutoff-independent contact interaction; taking $\Lambda\to\infty$ requires a separately defined renormalization prescription.

Convergence and failure modes

Vary quadrature order, interval mapping, and cutoff independently. A small linear-system residual establishes the discrete equation, not the continuum integral. Compare with radial partial-wave integration for a potential supported in both representations.

A finite $\eta>0$ in $E+i\eta$ is possible, but the grid must resolve its narrow Lorentzian before extrapolating $\eta\to0$. Taking $\eta$ smaller than the quadrature resolution can remove the imaginary contribution instead of approximating it.

For the finite discretized equation, a kernel spectral radius below one guarantees convergence of Born iteration. A direct linear solve can still succeed when that iteration fails: a large kernel eigenvalue need not make $I-K$ singular. The separable example below makes this distinction explicit.

Exercises

For the separable model, sum the Born series formally and identify its convergence condition.

Solution

The scalar series is $\tau=\lambda[1+\lambda I+(\lambda I)^2+\cdots]$, which converges when $|\lambda I|<1$ and sums to $\lambda/(1-\lambda I)$. The analytic solution remains meaningful outside that disk except at its poles, so direct iteration can fail where a linear solve succeeds.

References

• Descouvemont, Pierre, and Daniel Baye, “The R-matrix theory,” Reports on Progress in Physics 73, 036301 (2010), arXiv:1001.0678 — internal-region bases and channel matching.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.