Numerical study of the two-boson bound-state problem with and without partial-wave decomposition

TL;DR

This work develops and validates two complementary numerical formulations for the two-boson bound-state problem: a traditional 1D partial-wave Lippmann–Schwinger equation and a 2D vector-variable formulation that operates directly in momentum space. Using both a separable Yamaguchi potential and local Malfliet–Tjon interactions, the authors demonstrate exact numerical equivalence of the two approaches down to $10^{-10}$ MeV and derive analytic cut-off error expressions for the Yamaguchi case, providing rigorous benchmarks for discretization and truncation effects. The 2D method is shown to be essential for higher-energy applications and for extending to three- and four-body systems, while the Yamaguchi benchmarks enable precise error control and cross-checks with coordinate-space observables. Overall, the results establish a reliable reference standard for high-precision few-body codes and offer practical guidance for applying vector-variable formulations to complex few-body problems.

Abstract

The validation of numerical methods is a prerequisite for reliable few-body calculations, particularly when moving beyond standard partial-wave decompositions. In this work, we present a precision benchmark for the two-boson bound-state problem, solving it using two complementary formulations: the standard one-dimensional partial-wave Lippmann--Schwinger equation and a two-dimensional formulation based directly on vector variables. While the partial-wave approach is computationally efficient for low-energy bound states, the vector-variable formulation becomes essential for scattering applications at higher energies where the partial-wave expansion converges slowly. We demonstrate the high-precision numerical equivalence of both methods using rank-one separable Yamaguchi potentials and non-separable Malfliet--Tjon interactions. Furthermore, for the Yamaguchi potential, we derive exact analytical expressions quantifying the systematic errors introduced by finite momentum- and coordinate-space cut-offs. These analytical bounds provide a rigorous tool for disentangling discretization errors from truncation effects in few-body codes. The results establish a reliable reference standard for validating the vector-variable approaches essential for future three- and four-body calculations.

Figures (1)

• Figure 1: Convergence of the deviations $\Delta\langle H_0\rangle$ and $\Delta\langle V\rangle$ from their exact values as functions of the momentum cut-off $p_{\rm cut}$ (left) and the radial cut-off $r_{\rm cut}$ (right) for the YAMA-23 potential.