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

TL;DR

This paper benchmarks two numerical methods for the two-boson bound-state problem, demonstrating their equivalence and deriving error bounds, to improve reliability in few-body quantum calculations.

Contribution

It introduces a high-precision benchmark comparing partial-wave and vector-variable formulations, including analytical error estimates for systematic uncertainties.

Findings

Both methods are numerically equivalent for the tested potentials.

Analytical bounds for systematic errors are derived for Yamaguchi potentials.

The benchmark provides a baseline for future three- and four-body calculations.

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 highly controlled methodological benchmark that provides a detailed baseline for vector-variable algorithms intended for more complex three- and four-body calculations.