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

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.