This paper compares three different numerical ways to solve the same quantum bound-state problem for three identical bosons. The goal was to check whether methods that treat the particle momenta very differently give the same binding energies and spatial properties, and to measure the size of the remaining numerical errors.

The team ran a controlled set of calculations in momentum space. They compared a one-dimensional (1D) spectator-amplitude approach, a two-dimensional (2D) partial-wave approach, and a three-dimensional (3D) vector-variable approach. Partial waves mean expanding the angular dependence into a finite set of angular-momentum components; the 3D vector-variable method keeps the full momentum vectors. To make a fair test, the authors embedded the same finite partial-wave interaction space in each formulation so that any differences would come from numerical choices such as discretization, interpolation and quadrature (numerical integration).

The results show very tight agreement. For separable model interactions the binding energies from the three methods agree at the 10^(-6) MeV level. For local interactions the agreement is at the few ×10^(-6) to 10^(-5) MeV level. The authors also solved the 2D and 3D equations in two algebraic forms: a t-matrix-driven form (where the two-body transition matrix is used) and a bare-potential-driven form (where the interaction potential appears directly). The close match between these forms supports the correctness of the permutation handling, quadrature rules, and interpolation used in the codes.

As an independent check, the momentum-space wave functions were transformed to coordinate space using Fourier transforms. The coordinate-space results gave consistent norm decompositions (how probability distributes among components) and spatial observables. That agreement provides a second independent confirmation of the momentum-space solutions.