Numerical study of the two-boson bound-state problem with and withoutpartial-wave decomposition

The validation of numerical methods is a prerequisite for reliable few-bodycalculations, particularly when moving beyond standard partial-wavedecompositions. In this work, we present a precision benchmark for the two-bosonbound-state problem, solving it using two complementary formulations:the standard one-dimensional partial-wave Lippmann--Schwinger equation and atwo-dimensional formulation based directly on vector variables. While thepartial-wave approach is computationally efficient for low-energy boundstates, the vector-variable formulation becomes essential for scattering applicationsat higher energies where the partial-wave expansion converges slowly. Wedemonstrate the high-precision numerical equivalence of both methods usingrank-one separable Yamaguchi potentials and non-separable Malfliet--Tjoninteractions. Furthermore, for the Yamaguchi potential, we derive exactanalytical expressions quantifying the systematic errors introduced by finitemomentum- and coordinate-space cut-offs. These analytical bounds provide arigorous tool for disentangling discretization errors from truncation effectsin few-body codes. The results establish a reliable reference standard forvalidating the vector-variable approaches essential for future three- andfour-body calculations. PAC Codes: 21.45.-v , 03.65.Ge , 02.60.Nm , 03.65.Nk