SPARSE: Scattering Poles and Amplitudes from Radial Schrödinger Equations

TL;DR

SPARSE addresses the direct computation of scattering poles and amplitudes from a system of coupled radial Schrödinger equations. It advances a finite-difference, multi-channel approach that constructs a sparse, banded Hamiltonian, solves for numerical radial wavefunctions, and extracts a real, symmetric $K$-matrix from large-$r$ asymptotics. By combining analytic asymptotics, integrals to obtain transformation matrices, and AAA-based pole extraction, SPARSE yields resonances, couplings, and cross sections with high efficiency for tens of channels. The methodology enables parameter studies and data-driven fits without resorting to external approximations, and is demonstrated via a two-channel showcase with a bound-state induced resonance. The resulting pipeline—input via CSV, bound-state support, and tutorial notebooks—facilitates practical applications in nonrelativistic scattering problems.

Abstract

We introduce an algorithm for the solution of a large system of radial Schrödinger equations for scattering states. The system of differential equations is approximated as an ordinary linear nonhomogeneous system using the finite difference method. Dirichlet boundary conditions are imposed at the origin and at an arbitrary large radius. The $K$-matrix for physical energies is calculated from the numerical solutions of the system by comparison to the analytical real solutions at large distances. Scattering poles and amplitudes are calculated from the physical $K$-matrix.

Figures (2)

• Figure 1: Potential matrix elements for the SPARSE showcase example. Axes have arbitrary units.
• Figure 2: $K$-matrix (left) and square modulus of the $T$-matrix (right) in the SPARSE showcase example. The horizontal axes have arbitrary units.