Scattering and Resonance Computation

Scattering computations must represent an unbounded exterior while keeping the numerical problem finite. Different methods encode that exterior through matching, integral resolvents, transfer matrices, or non-Hermitian boundaries. This chapter explains their implementation and convergence; Scattering Theory explains the physical interpretation of amplitudes, flux, channels, and poles.

Enter this chapter

Begin with continuum states in finite boxes to distinguish discrete normalization from continuum strength densities. A finite Hermitian box has real levels; its apparent positive-energy states do not carry a resonance width.

For a short-range central potential, proceed through partial-wave numerics and stable phase extraction. The former propagates the regular solution, while the latter controls nodes, derivative conventions, phase branches, and threshold sensitivity.

Convert the resulting scattering matrix into observables with cross-section computation. Angular cutoff and forward-angle convergence remain necessary after the radial equation has converged.

Alternative stationary formulations

Lippmann–Schwinger numerics solves a momentum-space integral equation. Its decisive numerical issue is the on-shell pole and outgoing prescription. R-matrix methods solve an internal basis problem and pass boundary data to an exterior channel calculation.

Continuum transfer matrices are effective for piecewise one-dimensional potentials but become ill-conditioned in long evanescent regions. Lattice transfer matrices use stabilized products to extract Lyapunov exponents and localization lengths. Their matrix conventions and observables differ, so the two constructions should not be interchanged without derivation.

Outgoing boundaries and resonances

Complex absorbing potentials remove outgoing amplitude in a numerical layer. They require reflected-flux calibration over the momentum range of interest. Exterior complex scaling rotates the exterior coordinate and needs a correctly transformed operator and interface.

The complex-energy resonance page tracks candidates under those boundary changes and compares them with scattering phases or time-dependent decay. For $E_\star=E_r-i\Gamma/2$, probability decays as $e^{-\Gamma t/\hbar}$ only when a pole contribution dominates. Artificial absorber decay and discretized continuum modes must be excluded.

A diagnostic benchmark sequence

First set the potential to zero: phases vanish, reflection vanishes, and the elastic $S$ matrix is unity. Then use a hard-sphere $s$ wave, where $\delta_0=-kR$ modulo $\pi$, to test the phase sign. A finite square well tests interface matching and threshold scattering-length divergence. A square barrier tests exponentially small transmission and exposes unstable transfer products.

For an integral solver, a rank-one separable potential reduces the equation to a scalar denominator and gives an independent check of the on-shell imaginary term. For a resonance solver, vary angle or absorber parameters and the basis independently before interpreting an apparent stationary eigenvalue.

Convergence and physical checks

Record units, potential or lattice parameters, exterior conventions, normalization, boundary locations, discretization, solver residuals, and the variations used to establish convergence. Verify flux conservation when the model requires it and distinguish intentional absorption or inelastic loss.

Do not combine all refinements into one larger run. Separate interior resolution, exterior matching, angular cutoff, energy mesh, and pole conditioning so that the source of uncertainty can be identified. Agreement between two formulations of the same benchmark is especially useful because their dominant errors are different.

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.
• Johnson, B. R., “The multichannel log-derivative method for scattering calculations,” Journal of Computational Physics 13, 445–449 (1973), doi:10.1016/0021-9991(73)90049-1 — stable radial propagation.
• Moiseyev, Nimrod, “Quantum theory of resonances: calculating energies, widths and cross-sections by complex scaling,” Physics Reports 302, 212–293 (1998), doi:10.1016/S0370-1573(98)00002-7 — non-Hermitian resonance calculations.
• Taylor, John R., Scattering Theory: The Quantum Theory of Nonrelativistic Collisions, Dover Publications (2006; original edition 1972) — scattering normalization, amplitudes, and analytic structure.